Spurgeon...

J B Taylor explains that...

In its original context this reflected the Psalmist’s own jubilation at his vindication over the enemies who had rejected him, but in its liturgical setting in the Feast of Tabernacles the psalm came to refer more to national than to personal deliverance. In rabbinical exegesis it was accorded a Messianic interpretation and this prepared the way for its use by Christ of himself in Mt 21:42; Mark 12:10; Luke 20:17.  (Wood, D. R. W.. New Bible Dictionary InterVarsity Press)

The Greek word time reflects a manifestation of esteem (the regard with which one is held), honor, reverence. Time is a valuing by which the price is fixed or an estimation of the value of a thing. In the present context, time is descriptive of the inestimable, infinite worth of Christ our Rock.

In this passage Peter contrasts the significance of the Living Stone to believers with what this same Living Stone becomes to unbelievers who refuse to allow Him to become precious to them (by believing in Him).

G Campbell Morgan comments that...

The declaration is not that believers know the preciousness of Christ; it is rather that they share it.

The idea of preciousness is that of honour, and therefore of honourableness, that is, of the qualities that are worthy of honour. This is the thought of the statement, then. The qualities of Christ that create His preciousness, His honour, are placed at the disposal of the believer.

Twice already had the Apostle described the Lord as "precious" (see notes 1 Peter 2:4; 2:6). In both cases the description was a declaration of God's estimate of Him. He was the rejected of men, but with God He was elect, precious. We know the things in Christ which made Him precious, honourable, in the sight of God. They were the things of His purity, His love, His conformity to all the perfect will of God. Here, then, is the wonder of this declaration.

All these things are communicated to those who believe in Him. His very life 'and nature are given to the believer, and, by the might of their working, make that believer precious with His preciousness.

He is the living Stone, and those who come to Him, who believe in Him, receive that very quality of life which is His, and so they become living stones.

It is in the power of that preciousness that they become "an elect race, a royal priesthood, a holy nation, a people for God's own possession," and so are enabled "to show forth the excellencies" of God.

Spurgeon comments that...

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This is a text on which I think I could preach in my sleep. And I believe that if I were dying and were graciously led into the old track, I could with my last breath pour out a heart full of utterance on this delightful verse. I am sure it contains the marrow of what I have always taught in the pulpit.

Vincent notes that pisteuo...

means to persuade, to cause belief, to induce one to do something by persuading, and so runs into the meaning of to obey, properly as the result of persuasion

In secular Greek literature, as well as in the New Testament, pisteuo (pistis, pistos) has a basic meaning of an intellectual assent or a belief that something is true. Michel says that this use arose during the Hellenistic period. During the struggle with skepticism and atheism, it acquired the sense of conviction concerning the existence and activity of the Greek gods. Thayer calls this the intransitive use of the word which conveys the idea of to be sure or be persuaded that something is a fact. This kind of faith does not require any action on the part of the believer but only an intellectual acceptance. As discussed below, James used this type of faith as an example of a dead faith stating that "The devils also believe, and tremble" (James 2:19).

The other secular Greek meaning that is the more common use in the New Testament is the transitive or active use which means to "put faith in" or "rely upon" someone or something. Sometimes it has even stronger meaning: "To entrust something to another." In classical usage it denoted conduct that honored a previous agreement, such as the honoring of a truce between opposing armies (Iliad 2.124). The meaning of entrusting something to someone is found in Xenophon (Memorabilia 4.4.17). An example of this use in the New Testament is 2 Timothy 1:12. Paul said

I know whom I have believed, and am persuaded that he is able to keep that which I have committed unto him against that day (see note 2 Timothy 1:12) (Comment: Here pisteuo means to trust in or rely upon Christ to save us)

Pisteuo means to entrust oneself to an entity in complete confidence. To believe in with the implication of total commitment to the one who is trusted. As discussed below Christ is the object of this type of faith that relies on His power and nearness to help, in addition to being convinced that His revelations or disclosures are true.

The noun pistis and the verb pisteuo, mean an adherence to, committal to, faith in, reliance upon, trust in a person or an object, to be persuaded of or convinced of something, to place one's confidence in, to trust.

Pisteuo can also mean to be confident about or to be firmly persuaded as to something, and so Paul writes...

One man has faith (pisteuo) that he may eat all things, but he who is weak eats vegetables only. (see note Romans 14:2) (Here the believing conveys the sense of having an opinion, thinking)

As noted above, pisteuo can refer to an heart belief (saving faith, genuine belief that leads to salvation, this believing involves not only the consent of the mind, but an act of the heart and will of the subject) or an intellectual belief (mental assent, "head" knowledge, not associated with bringing salvation if it is by itself), both uses demonstrated by Jesus statement in John 11, 

John 11:26 Everyone who lives and believes (refers to genuine saving faith) in Me shall never die. Do you believe (intellectually) this?

James 2:19 You believe (pisteuo) that God is one. You do well; the demons also believe (pisteuo), and shudder.

Comment: In this passage, James explains that not all believing will result in salvation. The believing he is describing in this passage is a mental or intellectual believing that is not associated in a change in one's heart and thus in one's behavior or actions. Belief in the New Testament sense that effects the new birth denotes more than a "demonic" like, intellectual assent to a set of facts or truths. The demons believe but they are clearly not saved. Genuine belief does involve an intellectual assent and consent of one's mind, but also includes an act of one's heart and will. Biblical saving faith is not passive assent but an active staking of one's life on the claims of God. The respected Greek lexicon author W E Vine defines belief as consisting of

(1) a firm conviction which produces full acknowledgment of God's revelation of Truth - (2Thes 2:11 -"in order that they all may be judged who did not believe [pisteuo] the truth, but took pleasure in wickedness.")

(2) a personal surrender to the Truth (Jn 1:12 "But as many as received Him, to them He gave the right to become children of God, even to those who believe [pisteuo] in His name") and

(3) a conduct inspired by and consistent with that surrender.

Pisteuo is found 24 times in the Septuagint (LXX) and the first use by Moses is one of the most important uses of pisteuo in all of Scripture...

Genesis 15:6 Then he (Abraham) believed (Hebrew = 'āman; LXX = pisteuo) in the LORD; and He reckoned it to him as righteousness. (Comment: Note that in the OT, salvation was by faith, not works. Paul explains that Abraham heard the gospel - see Galatians 3:8. It is also worth noting that the Hebrew word for "believe" in this verse is 'āman means to confirm, support or uphold and conveys the essential idea that one remains steadfast. At the heart of the meaning of the root of the Hebrew verb 'āman is the idea of certainty or firmness. The derivatives reflect the concept of certainty and dependability. In other words faith is not a blind leap into the dark but a confident commitment to the One about Whom abundant evidence bears ample testimony of His eternal, immutable trustworthiness. Faith is far more than mere hope that something unlikely may happen. It is a deep, internal certainty, rooted in our trust of what God has said.)

As alluded to above, Biblical faith or believing is not synonymous with mental assent alone, which is not genuine (saving) faith. For example, the apostle John distinguishes two types of believing using the verb pisteuo, one of which is only a superficial profession...

John 2:22 When therefore He was raised from the dead, His disciples remembered that He said this; and they believed (pisteuo) the Scripture and the word which Jesus had spoken. (Morris in Defenders Study Bible writes "Note the superior category of faith of the disciples to that of the "many" in John 2:23 who believed "when they saw the miracles," but soon fell away. The disciples did not believe because of the miracles but because of the Scripture and Jesus' words. It is far better to place one's faith in God's Word than in signs and wonders.")
23 Now when He was in Jerusalem at the Passover, during the feast, many believed (pisteuo) in His name, beholding His signs which He was doing. (Note that their belief was associated with His signs)
24 But Jesus, on His part, was not entrusting (pisteuo) Himself to them, for He knew all men (Morris writes "Although many in the Jerusalem crowd "believed in his name when they saw the miracles" (John 2:23), Jesus did not "believe" in them because He knew their hearts and knew their outward faith in Him was only superficial)
25 and because He did not need anyone to bear witness concerning man for He Himself knew what was in man. (The Ryrie Study Bible notes that "The contrast is between people who put their trust (pisteuo) in Jesus, and Jesus, who does not put His trust in people because He knows their motives and thoughts. Enthusiasm for the spectacular is present in them, but Jesus looks for genuine faith.) (John 2:22-25)

In another example of believing that falls short of genuine saving belief John records that when Jesus spoke to the Jews "who had believed (pisteuo) Him" (John 8:31) but as their subsequent actions demonstrated their belief was not genuine for Jesus accused them declaring "you are seeking to kill Me" (John 8:40) and after several heated exchanges, these same "believing" Jews "fulfilled prophecy" and indeed sought to kill Jesus, picking

up stones to throw at Him; but Jesus hid Himself, and went out of the temple. (John 8:59) (Comment: These Jews had a profession but not genuine possession in respect to their belief in Jesus).

Wuest writes that when pisteuo refers...

to the faith which a lost sinner must place in the Lord Jesus in order to be saved, they include the following ideas; the act of considering the Lord Jesus worthy of trust as to His character and motives, the act of placing confidence in His ability to do just what He says He will do, the act of entrusting the salvation of his soul into the hands of the Lord Jesus, the act of committing the work of saving his soul to the care of the Lord. This means a definite taking of one’s self out of one’s own keeping and entrusting one’s self into the keeping of the Lord Jesus. (Wuest, K. S. Wuest's Word Studies from the Greek New Testament: Eerdmans or Logos)

Nelson's New Illustrated Bible Dictionary says that...

A belief that saves is one that rests in the finished work of Christ; it trusts God alone for salvation (John 3:16). Believers are those who have trusted God with their will as well as their mind (see notes Romans 1:16; Romans 3:22; 1Thessalonians 1:7). (Youngblood, R. F., Bruce, F. F., Harrison, R. K., & Thomas Nelson Publishers. Nelson's New Illustrated Bible Dictionary)

Larry Richards has an excellent discussion on believing writing  that...

Originally this word group (pisteuo, pistis, pistos) seems linked with a more formal contract between partners. It stressed faithfulness to the agreement made or trustworthiness in keeping promises. In time the use expanded. In the classical period, writers spoke of trust in the gods as well as trust in people. In the Hellenic era, "faith in God" came to mean theoretical conviction about a particular doctrine, a conviction expressed in one's way of life. As different schools of philosophy and religion developed, the particular emphasis given pistis was shaped by the tradition within which it was used. The NT retains the range of meanings. But those meanings are refined and reshaped by the dynamic message of the gospel.

The verb (pisteuo) and noun (pistis) are also used with a number of prepositions. "To believe through" (dia) indicates the way by which a person comes to faith (Jn 1:7; 1 Peter 1:21 [note]). "Faith en" indicates the realm in which faith operates (see notes Ephesians 1:15; Colossians 1:4; 2 Timothy 3:15). The most important construction is unique to the NT, an invention of the early church that expresses the inmost secret of our faith. That construction links faith with the preposition eis, "to" or "into." This is never done in secular Greek. In the NT it portrays a person committing himself or herself totally to the person of Jesus Christ, for our faith is into Jesus. (Ed note:  Leon Morris in "The Gospel According to John" agrees with Richards writing that "Faith, for John, is an activity which takes men right out of themselves and makes them one with Christ" indicating that Morris likewise understands the Greek preposition eis in the phrase pisteuo eis, to be a significant indication that NT faith is not just intellectual assent but includes a "moral element of personal trust.")

One other aspect of the NT's use of faith words is fascinating. Usually the object of faith is Jesus. Only twelve verses have God as the object of faith (Jn 12:44; 14:1; Acts 16:34; see notes Romans 4:3, 4:5, 4:17, 4:24; Gal 3:6; 1Thessalonians 1:8 [note]; Titus 3:8 [note]; Hebrews 6:1 [note]; 1Peter 1:21 [note]). Why? The reason is clearly expressed by Jesus himself: "I am the way and the truth and the life. No one comes to the father except through me" (Jn 14:6). God the Father has revealed himself in the Son. The Father has set Jesus before us as the one to whom we must entrust ourselves for salvation. It is Jesus who is the focus of Christian faith. (Richards, L O: Expository Dictionary of Bible Words: Regency - a highly recommended resource)

J. B. Lightfoot discusses the concept of faith in his commentary on Galatians. He notes that in Hebrew, Greek, and Latin, the definition of the word for faith

hovers between two meanings: trustfulness, the frame of mind which relies on another; and trustworthiness, the frame of mind which can be relied upon...the senses will at times be so blended together that they can only be separated by some arbitrary distinction. The loss in grammatical precision is often more than compensated by the gain in theological depth...They who have faith in God are steadfast and immovable in the path of duty.

Faith, like grace, is not static. Saving faith is more than just understanding the facts and mentally acquiescing. It is inseparable from repentance, surrender, and a supernatural longing to obey. None of those responses can be classified exclusively as a human work, any more than believing itself is solely a human effort.

Faith is manifest by not believing in spite of evidence but obeying in spite of consequence. John uses pisteuo to demonstrate the relationship between genuine faith and obedience writing...

He who believes (pisteuo - present tense = continuous) in the Son has eternal life but he who does not obey (apeitho - present tense = continuously disobey, habitually, as their lifestyle)  the Son shall not see life, but the wrath of God abides on him. (John 3:36) (Comment: The verb apeitho conveys more an attitude of unbelief but also involves deliberate disobedience, conscious resistance to authority)

Charles Swindoll commenting on faith and obedience in John 3:36 concludes that...

In John 3:36 the one who "believes in the Son has eternal life" as a present possession. But the one who "does not obey the Son shall not see life." To disbelieve Christ is to disobey Him. And logically, to believe in Christ is to obey Him. As I have noted elsewhere, "This verse clearly indicates that belief is not a matter of passive opinion, but decisive and obedient action." (quoting J. Carl Laney)...Tragically many people are convinced that it doesn't really matter what you believe, so long as you are sincere. This reminds me of a Peanuts cartoon in which Charlie Brown is returning from a disastrous baseball game. The caption read, "174 to nothing! How could we lose when we were so sincere?" The reality is, Charlie Brown, that it takes more than sincerity to win the game of life. Many people are sincere about their beliefs, but they are sincerely wrong!" (Swindoll, C. R., & Zuck, R. B. Understanding Christian Theology.: Thomas Nelson Publishers or Logos Bible Software) (This book is recommended if you are looking for a very readable, non-compromising work on "systematic theology". Wayne Grudem's work noted above is comparable.)

Subjectively faith is firm persuasion, conviction, belief in the truth, veracity, reality or faithfulness (though rare). Objectively faith is that which is believed (usually designated as "the faith"), doctrine, the received articles of faith. Click  separate study of "the faith (pistis)"

True faith is not based on empirical evidence but on divine assurance.

When missionary John Paton  was translating the Scripture for the South Sea islanders, he was unable to find a word in their vocabulary for the concept of believing, trusting, or having faith. He had no idea how he would convey that to them. One day while he was in his hut translating, a native came running up the stairs into Paton's study and flopped in a chair, exhausted. He said to Paton,

It's so good to rest my whole weight in this chair.

John Paton had his word: Faith is resting your whole weight on God. That word went into the translation of their New Testament and helped bring that civilization of natives to Christ. Believing is putting your whole weight on God. If God said it, then it's true, and we're to believe it.

Nothing before, nothing behind,
The steps of faith
Fall on the seeming void, and find
The rock beneath -- Whittier

As the great British preacher C H Spurgeon said...

Faith is the foot of the soul by which it can march along the road of the commandments.

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The great Baptist preacher Adrian Rogers...

Corrie ten Boom...

Faith, mighty faith, the promise sees,
And looks to that alone;
Laughs at impossibilities,
And cries it shall be done. -- Charles Wesley

The great American evangelist, D L Moody...

Spurgeon in his devotional Morning and Evening has the following devotional thoughts on this verse...

As all the rivers run into the sea, so all delights centre in our Beloved. The glances of His eyes outshine the sun: the beauties of His face are fairer than the choicest flowers: no fragrance is like the breath of His mouth. Gems of the mine, and pearls from the sea, are worthless things when measured by His preciousness.

Disbelieve (569) (apisteo from a = without + pistos = believing, faithful) means literally without believing. They refuse to believe and thus are unfaithful. To disbelieve, to doubt or not to acknowledge. To betray a trust. Unbelief is a failure to respond to God with trust (pistis) and at heart shows, not doubt, but rejection.

Apisteo is used 6 times in the NAS: (Mark 16:11, Mark 16:16, Luke 24:11, 41 - these all refer to disbelieving Christ's resurrection. Apisteo is used to describe the Jews who are listening to Paul's testimony of Christ in Acts 28:24. See notes on the other two uses in Romans 3:3, 2 Timothy 2:13)

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Disobedient (KJV) (544) (apeitheo from a = without + peítho = persuade) literally describes one who refuses to be persuaded and who disbelieves willfully and perversely. Apeitheo in the present context  means that these individuals possess an attitude of unbelief because they deliberately disobey, consciously resist and rebel against authority and finally manifest an obstinate rejection of the will (truth) of God. The present tense indicates that this is their lifestyle or their habitual practice. They live in continual disobedience to the Almighty, Holy God. To be sure, we all disobey from time to time. That is not what Peter is referring to here. Instead he is describing the individual with an unregenerate heart who habitually, continually disobeys (as a lifestyle) what he or she knows to be the truth.

Unbelievers were constantly rejecting, ridiculing, mocking, abusing, threatening, and persecuting him—even while he was sharing the glorious news of eternal life with them.

Marvin Vincent in discussing apeitheo in John 3:36 writes that..

Disbelief is regarded in its active manifestation, disobedience. The verb peitho means to persuade, to cause belief, to induce one to do something by persuading, and so runs into the meaning of to obey, properly as the result of persuasion...Obedience, however, includes faith. (Ed Note: See discussion of phrase obedience of faith at Romans 1:5). (Vincent, M. R. Word studies in the New Testament Vol. 2, Page 1-109)