For the appeal we make does not spring from error (NIV)

For (1063)(gar) is a conjunction which introduces an explanation and here indicates the basis for their boldness.

Hiebert adds that...

With his explanatory "for" Paul indicates what enabled and obligated the missionaries to preach with boldness in spite of suffering and opposition. He categorically denies three false motives imputed to them (1Thes 2:3) and asserts that their preaching was that of men who had been tested by God and divinely entrusted with the gospel (1Thes 2:4). Had they acted under the false motives attributed to them they would have been unable to preach as they did under the circumstances. (Hiebert, D. Edmond: 1 & 2 Thessalonians: BMH Book. 1996)

One thought widely held by many conservative commentators is that in this verse Paul explains that he, Silas and Timothy are innocent of the malicious charges made against them by the jealous Jews.

F F Bruce notes that there were...

 So many wandering charlatans (γόητες) made their way about the Greek world, peddling their religious or philosophical nostrums, and living at the expense of their devotees (like Lucian’s false prophet Alexander), that it was necessary for Paul and his friends to emphasize the purity of their motives and actions by contrast with these. The same kind of contrast is made by Dio Chrysostom in Oration 32 (mentioned in comment on v 2 above). “The normal heathen ‘missionaries’ … were itinerant apostles and miracle-workers of the most varied persuasions, heralds of heathen gods, and dispensers of salvation, adroit and eloquent, ardent and evoking ardor, but also smart and conceited in extolling the mighty acts of their gods and fooling the masses … There can be no doubt that they constituted dangerous rivals of the gospel and that general popular opinion expected the Christian missionaries to be able to vie with them” (Bornkamm, Paul, 64). (Bruce, F F: 1 and 2 Thessalonians. Word Biblical Commentary. Dallas: Word, Incorporated. 1982 or Logos)

In his second letter to the Thessalonians Paul makes allusion to a deceptive message writing...

Now we request you, brethren, with regard to the coming of our Lord Jesus Christ, and our gathering together to Him, 2 that you may not be quickly shaken from your composure or be disturbed either by a spirit or a message or a letter as if from us, to the effect that the day of the Lord has come. 3 Let no one in any way deceive you, for it will not come unless the apostasy comes first, and the man of lawlessness is revealed, the son of destruction, 4 who opposes and exalts himself above every so-called god or object of worship, so that he takes his seat in the temple of God, displaying himself as being God. (2Thess 2:1-4)

Not (3756) (ouk) is an adverb strongly negating an alleged fact.

MacDonald nicely summarizes this verse commenting that...

The apostle’s exhortation to believe the gospel was true in its source, pure in its motive, and dependable in its method. As to its source, it did not spring from false doctrine but from the truth of God. As to its motive, the apostle looked on the Thessalonians unselfishly, with their good in view, and not with any ulterior, impure desire. As to its method, there was no clever plot to deceive them. Apparently his jealous enemies were accusing him of heresy, lustful desire, and craftiness. (MacDonald, W & Farstad, A. Believer's Bible Commentary: Thomas Nelson or Logos)

Ryrie writes that...

Paul attacks what must have been charges brought against him: of error (i.e., that the gospel he preached was based on error); of impurity (that Christianity encouraged sexual immorality); and of deceit (that his methods were underhanded). (The Ryrie Study Bible: New American Standard Translation: 1995. Moody Publishers)

Exhortation (3874) (paraklesis from pará = side of, beside + kaléo = call) literally means a calling to one's side for admonition, encouragement, exhortation, consolation or comfort.

Gloag has an interesting note on paraklesis in the present context writing that...

This word has a twofold signification, denoting both “exhortation” and “consolation;” when it refers to the moral conduct it denotes exhortation, but when it is an address to a sufferer it denotes consolation. In the gospel these two meanings are blended together. (The Pulpit Commentary: New Testament; Old Testament; Ages Software  or Logos)

Barnes explains exhortation refers to their

exhortation to embrace the gospel. The word seems to be used here so as to include preaching in general. The sense is, that the means which they used to induce them to become Christians were not such as to delude them.  (Albert Barnes. Barnes NT Commentary).

Green adds that...

The gospel is not simply an announcement of the good news but also a call to respond to the divine initiative contained in it. (Pillar NT Commentary. Logos)

Alexander remarks

As addressed to the careless, slothful, tempted, fallen, it is exhortation; as addressed to the sad and seeking it is solace and comfort.

MacArthur says that in the present context it

means an urgent cry, appeal, or call, with an emphasis on judgment. Such usage stressed for Paul’s readers the urgency and directness of his preaching. (MacArthur, John: 1 & 2 Thessalonians. Moody Press or Logos)

What exhortation did Paul present when he came to Thessalonica?

Now when they had traveled through Amphipolis and Apollonia, they came to Thessalonica, where there was a synagogue of the Jews. And according to Paul's custom, he went to them, and for three Sabbaths reasoned with them from the Scriptures (Old Testament teachings about Messiah), explaining ("opening them") and giving evidence that the Christ (the Messiah) had to suffer and rise again from the dead, and saying, "This Jesus whom I am proclaiming to you is the Christ (the Messiah)." (Note: that here Paul proclaimed the essential Gospel points - Messiah died, implied that he was buried, was resurrected - see notes on 1Corinthians 15:1; 15:1; 15:2; 15:3; 15:4; 15:5; 15:6; 15:8) 4 And some of them were persuaded and joined Paul and Silas, along with a great multitude of the God-fearing Greeks and a number of the leading women. 5 But the Jews, becoming jealous and taking along some wicked men from the market place, formed a mob and set the city in an uproar; and coming upon the house of Jason, they were seeking to bring them out to the people. (Acts 17:1-5)

Paraklesis in this context refers to persuasive discourse,  stirring address—instructive, admonitory, consolatory; powerful hortatory discourse.

Paraklesis implies an appeal having for its object the direct benefit of those addressed, which may be either hortatory or conciliatory depending on circumstances. The word was used to encourage soldiers before going into battle and it was said that encouragement was necessary for hired soldiers; but for those who fight for life and country no exhortation is required

The word appeal has been adopted for many translations (TCNT, Weymouth, Moffatt, Goodspeed, RSV, NEB, NIV).

TDNT on paraklesis and parakaleo...writes that these two words have a wide range of meaning...

1. the first sense being that of calling to either literally or with such nuances as calling for aid, inviting, and summoning.
2. A second sense is that of beseeching, e.g., calling on the gods in prayer, or, from a superior to an inferior, proposing.
3. A third sense is that of exhorting or encouraging, and even on occasion of ?winning over? for a plan.
4. The final sense is that of comforting, mostly in the form of giving exhortation or encouragement in times of sorrow. This is not a common use. (Kittel, G., Friedrich, G., & Bromiley, G. W.  Theological Dictionary of the New Testament. Eerdmans)

Not come from error - In the first century world Paul lived in, there were many competing religions (not to different from our modern times!), and many ministers of those religions were motivated by greed and gain (certainly not different from today!). Accordingly, certain questionable practices common among unsound religious leaders found no place in his evangelistic labours. And so Paul begins by emphasizing that their exhortation was without any evil intent.

Neil notes that...

There has probably never been such a variety of religious cults and philosophic systems as in Paul’s day . . . ‘Holy men’ of all creeds and countries, popular philosophers, magicians, astrologers, crack-pots, and cranks; the sincere and the spurious, the righteous and the rogue, swindlers and saints, jostled and clamoured for the attention of the believing and the skeptical.

Error (4106) (plane) is an interesting word which describes a going astray, a straying about, a roaming or wandering, a forsaking of the right path. By straying about one is led away from the right way and thus roams here and there, which is not good in spiritual matters.

Plane is used figuratively in the NT of mental straying from the truth and thus refers to error, wrong opinion relate to morals or religion, a delusion or a deception. Plane thus describes a straying from orthodoxy or piety. The opposite of plane (error) is aletheia (truth). 

I agree with Richison that...

A plethora of people peddle their own viewpoints in the church today. Paul did not come with his own viewpoint; he came with the Word of God. We must set forth truth as it is rather than the way we want it to be. We should not manipulate Scripture for the sake of elegance in preaching. We simply set forth truth as it is. Our preaching should be as accurate as the multiplication table. We must never alter or water down the Word or the gospel. Inaccurately stating of God’s Word will ruin its message. (Ref)

In the NT plane is used only as a figurative extension of the primary meaning of roaming or wandering, whether in doctrine, 2 Peter 3:17 (note); 1 John 4:6, or in morals, Romans 1:27 (note); 2 Peter 2:18 (note); Jude 1:11, though in Scripture doctrine and morals are never really divided for one always behaves as one believes (or at least this should be the case)!

Herodotus (1, 30) writes of of Solon, who roamed (plane) the earth in search of new information -- to reiterate plane thus means a wandering from the path of truth.

In the present context, Paul was confident that their gospel message was genuine and stood in marked contrast to the erroneous teaching of their adversaries. Paul, Silas and Timothy had themselves not been carried away by any wiles of error, nor had they sought to mislead others by such winds of doctrine (see note Ephesians 4:14). Furthermore, they had not been deceived and were not deceivers (See note 2 Timothy 3:13, where related verb planao is used).

TDNT writes that the basic idea behind this word group (noun = plane, verb = planao)...

is that of going astray. Wandering is usually denoted, but the group may also be used for the pulsing of the blood through the body and the journeys of merchants. We find wanderers in tragedy, e.g., Io and Oedipus. In the case of Io the lack of a goal has an inner effect in a combination of geographical and spiritual wandering. (Kittel, G., Friedrich, G., & Bromiley, G. W.  Theological Dictionary of the New Testament. Eerdmans)

Vine writes plane is a...

wandering, whereby those who are led astray roam hither and thither, is always used in the N.T., of mental straying, wrong opinion, error in morals or religion....Errors in doctrine are not infrequently the effect of relaxed morality, and vice versa.

Vine goes on to add that...

The missionaries were confident theirs was a genuine gospel, standing out in contrast to the fatal delusion, for instance of which the apostle had already forewarned them, 2 Thessalonians 2:5. They had not themselves been carried away by any wiles of error, neither had they sought to mislead others by such wiles, Ephesians 4:14 (note); they had not been deceived, neither were they deceivers, see 2 Timothy 3:13 (note), where the corresponding verb, planao, is used (Vine, W. Collected writings of W. E. Vine. Nashville: Thomas Nelson or Logos)

OR IMPURITY: oude ex akatharsias:

Literally but not (3761) (oude) which is an absolute negation. This adverb proceeds impurity and deceit.

Impurity - is literally "absolutely never out of impurity". Although the Greek word akatharsia often refers to sexual misconduct as discussed below and this could be the primary meaning, the context refers primarily to their message and motives -- their preaching is not from impure motives such as ambition, pride, greed, popularity. Their message was not that which characterized the traveling charlatans of Paul's day.

Bruce explains that...

So many wandering charlatans made their way about the Greek world, peddling their religious or philosophical nostrums, and living at the expense of their devotees (like Lucian's false prophet Alexander), that it was necessary for Paul and his friends to emphasize the purity of their motives and actions by contrast with these.  (Bruce, F F: 1 and 2 Thessalonians. Word Biblical Commentary. Dallas: Word, Incorporated. 1982 or Logos) (Bolding added)

Impurity (167) (akatharsia from a = without + kathaíro = cleanse) is literally worthless material, waste; of graves decayed flesh, causing ceremonial uncleanness. It is a broad term referring to moral uncleanness in thought, word, and deed and thus describes a state of moral impurity and especially sexual sin or immorality (“the impurity of lustful, luxurious, profligate living”). It is not surprising that akatharsia was used to describe filth or refuse. Akatharsia medically referred to an infected, oozing wound and the general term often used of decaying matter, like the contents of a grave.  It defines a state of moral impurity, esp sexual sin, impurity, immorality, filthiness, state of moral impurity, especially in relationship to sexual sin (note Romans 1:24). It is a filthiness of heart and mind that defiles (mars, sullies, spoils) the person. The unclean person sees dirt in everything. The word akatharsia suggests especially that it defiles its participants, making them unusable for sacred purpose. While akatharsia includes sexual sin, it comes from a wider Septuagint (Greek translation of the Hebrew OT) usage where “unclean” could refer to anything that made a person unfit to go to the temple and appear before God. In a medical sense Hippocrates used this word to describe an infected, oozing wound with pus and crusty impurities that gather around the sore or wound.

What is “impure” is filthy and repulsive, especially to God, whether it is lewd actions or as in this verse preaching from impure motives.

MacArthur feels that impurity refers not just to their motives but actual practice writing that...

In Paul’s day many of the mystery religions and Greek cults practiced and even exalted sexual perversion. Those religions were very popular because in most of them the primary religious experience centered on the cult adherents having sex with a ritual temple prostitute or the cult leader. Temple orgies were not uncommon. Sexual intercourse had such a central role in those pagan religions because the members believed that when one had sex with a male leader or female prostitute—those supposedly closest to the gods—the individual connected with the deities. Therefore, through fornication they supposedly achieved some sort of mystical or metaphysical union with the gods. Thus wicked, unscrupulous leaders would seek converts for the purpose of having a sexual encounter with them. (MacArthur, John: 1 & 2 Thessalonians. Moody Press or Logos)

Vine commenting on the use here in 1Thessalonians makes note of...

the description of the false teachers in 2 Peter 2:18 (note), where sensuality and error are again associated. See also Jude 4, and Revelation 2:20 (note). Both in Corinth and in Thessalonica gross vice was consecrated to the service of religion, cp. Romans 1:22-32. (Vine, W. Collected writings of W. E. Vine. Nashville: Thomas Nelson or Logos)

Barnes comments that not...impurity means...

Not such as to lead to an impure life. It was such as to lead to holiness and purity. The apostle appeals to what they knew to be the tendency of his doctrine as an evidence that it was true. Most of the teaching of the heathen philosophers led to a life of licentiousness and corruption. The tendency of the gospel was just the reverse. (Albert Barnes. Barnes NT Commentary)

OR BY WAY OF DECEIT: oude en dolo:

By way of deceit - This is more literally never in deceit or guile. Paul is keenly sensitive against charges against the correctness of his message and the purity of his life. His message was not given so as to bait or trap his listeners, which is the meaning of dolos (discussed below).

Wiersbe explains that...

The word translated “guile” (deceit) carries the idea of “baiting a hook.” In other words, Paul did not trap people into being saved, the way a clever salesman traps people into buying his product. Spiritual witnessing and “Christian salesmanship” are different. Salvation does not lie at the end of a clever argument or a subtle presentation. Often we hear, “I don’t care what your method is, just so long as your message is right.” But some methods are unworthy of the Gospel. They are cheap, whereas the Gospel is a costly message that required the death of God’s only Son. They are worldly and man centered, whereas the Gospel is a divine message centered in God’s glory. (Wiersbe, W: Bible Exposition Commentary. 1989. Victor or Logos)

Deceit (1388) (dolos derived from dello meaning to bait) literally refers to a fishhook, trap, or trick all of which are various forms of deception. We use guile when we fish by using bait that appears like food to the fish. It is not food but a decoy. Dolos is a deliberate attempt to mislead, trick, snare or "bait" (baiting the trap in attempt to "catch" the unwary victim) other people by telling lies. It is a desire to gain advantage or preserve position by deceiving others. A modern term in advertising is called "bait and switch" where the unwary consumer is lured in by what looks like an price too good to be true!

Richison writes that...

Secular literature used “deceit” for a huckster. They used this word for a tavern keeper of the ancient world who would water down the wine of an inebriated person. There are those who also water down the Word. They use guile and tricks with the message of Christ. These are spiritual hucksters. Authenticity in content and delivery are the essence of the gospel. (Ref)

Dolos describes deceit, treachery, fraud, deceitful cunning, duplicity, trick, wile, guile, subtilty, snare, deliberate dishonesty. It represents a deliberate attempt to mislead other people by telling lies. It is a desire to gain advantage or preserve position by deceiving others. As noted the idea is to set bait so as to catch them. To bait or deceive in order to achieve one’s end. When a person wants something, he looks at the other person’s weakness or ignorance, and he tries to appeal to it. He appeals to it by deceiving and beguiling the person.

Vine writes that...

the preceding words deny a wrong source (error) and a wrong motive (impurity); deceit denies a wrong method. The meaning of the word is best seen in its first New Testament occurrence, Matthew 26:4 ("and they plotted together to seize Jesus by stealth {dolos} and kill Him"), or from the use of the corresponding verb in 2 Corinthians 4:2, “not handling the word of God deceitfully {doloo},” or craftily, that is, with some personal end in view. Such a charge was afterwards made against Paul at Corinth, 2 Corinthians 12:16, “being crafty I caught you with guile” words quoted, apparently, from the language of his detractors. (Vine, W. Collected writings of W. E. Vine. Nashville: Thomas Nelson or Logos)

Larry Richards explains that dolos...

Barclay amplifies the meaning of dolos noting that...