First, let's look at the three verses in question (all citations from Matthew 18 will be in purple):
"If your brother sins, go and show him his fault in private; if he listens to you, you have won your brother. But if he does not listen to you, take one or two more with you, so that by the mouth of two or three witnesses every fact may be confirmed. If he refuses to listen to them, tell it to the church; and if he refuses to listen even to the church, let him be to you as a Gentile and a tax collector." [Matthew 18:15-17; NASB]Now, let's examine these three verses bit by bit.
"If your brother sins, go and show him his fault in private; if he listens to you, you have won your brother." [v. 15]Many not used to the NASB translation may notice that something appears missing from this verse; the traditional reading of the verse is "if your brother sins against you." These two words have been used by some people to try to argue that Matthew 18:15 is solely dealing with personal affronts, and hence people who hear a Christian has sinned against someone else should, in essence, "mind their own business." The reason for this difference between translations is that the words "against you" are actually a textual variant. The NET translation notes for this verse read:
The earliest and best witnesses lack “against you” after “if your brother sins.” It is quite possible that the shorter reading in these witnesses (א B, as well as 0281 f1 579 pc sa) occurred when scribes either intentionally changed the text (to make it more universal in application) or unintentionally changed the text (owing to the similar sound of the end of the verb ἁμαρτήσῃ [hamartēsē] and the prepositional phrase εἰς σέ [eis se]). However, if the mss were normally copied by sight rather than by sound, especially in the early centuries of Christianity, such an unintentional change is not as likely for these mss. And since scribes normally added material rather than deleted it for intentional changes, on balance, the shorter reading appears to be original. NA27 includes the words in brackets, indicating doubts as to their authenticity.If "against you" is not to be considered part of the original manuscripts, then this signifies that this verse does not refer to a personal affront, and hence such a situation does not need to occur for us to confront a brother who sins. This also makes Matthew 18:15 perfectly in line with Galatians 6:1, where the apostle Paul writes "if anyone is caught in any trespass, you who are spiritual, restore such a one in a spirit of gentleness; each one looking to yourself, so that you too will not be tempted."
If a brother is discovered or sin, or we somehow discover that a brother has sinned, we are told to "show him his fault in private." The Greek for "show him his fault" is literally in the Greek "go reprove him" (ὕπαγε ἔλεγξον αὐτὸν). The word for "reprove" is the same used by the apostle Paul in 2 Timothy 4:2; the general idea is that you should go and demonstrate the guilt of your brother. Obviously this should be done, as the apostle Paul said, "in a spirit of gentleness," hoping to bring the brother to repentance and not simply make them feel guilty for their transgressions (though, if they are truly repentant, guilt should be a definite part of it). The goal of this confrontation is that the brother "listens to you" - that is, he repents of his sins, confesses them, and seeks to show the fruits of this repentance.
Some have argued that there should be given "time to repent." However, this is nowhere in the passage, and tends to be argued simply to give the person an excuse to continue in sin.
We are told to do this "alone" (μόνου). Much has been attempted to be said and done about this single word, and so let's understand two things about it:
What this does mean is that we should do this reproving in private, to respect the dignity of our brother. We should not bring it out into the open, or before the entire church, simply at the drop of a hat, unless there is some immediate danger or concern for another person's life or well-being.
What this doesn't mean is that we are absolutely forbidden from seeking advice or help on this matter. The "alone" here (as Mister Liantonio rightfully points out) simply contrasts it with the reproof involving one or two more witnesses in verse 16 - it is about the confrontation itself, not the prayerful consideration leading up to the confrontation.
Let us say, for the sake of example, that someone sees one of their fellow church members at a restaurant with a woman not his wife. Maybe he doesn't feel comfortable confronting the individual, but he goes to a friend of the man and says, "Hey, could you talk to your friend? I've seen him in sin." In this case, the friend would be taking over the regulations of Matthew 18:15. He may also make it more anonymous, but ask some brothers, "Hey, I've seen someone at my church in sin - should I confront him?" It is perfectly possible to seek godly council and not enter into the realm of gossip.
Of course, gossip should be avoided. However, there is a difference between "I've seen so-and-so committing sin, what should I do?" and "Hey, have you heard about what so-and-so did?"
"But if he does not listen to you, take one or two more with you, so that by the mouth of two or three witnesses every fact may be confirmed." [v. 16]Verse 16 presumes that, after the private confrontation, the brother in Christ did not repent, and has stubbornly continued without repentance, or have even remained active in sin. At this point, the person doing the confronting takes one or two more people who are also aware of the brother's transgression.
The reference to the "mouth of two or three witnesses" is actually a citation from the Law (Deu 19:15). This is also, incidentally, the "two or three" Christ is referring to in v. 20, which is perhaps one of the most misused passages of scripture: Christ is not referring to church fellowship, but to church discipline.
"If he refuses to listen to them, tell it to the church; and if he refuses to listen even to the church, let him be to you as a Gentile and a tax collector." [v. 17]Verse 17 assumes that the individual has not repented, even though he was confronted by two or three people on their sin. In this case, the person is brought forward to "the church." This suggests that the sin is to be handled in a local context, within the sinning individual's church, and by the church body.
The idea of being "as a Gentile and a tax collector" is that you should not associate with such a person on a high level (in the first century Jewish context, Gentiles were not allowed within the inner parts of the Temple, and tax collectors were considered among the very worst of sinners). Most would rightly assume that this means excommunication, or at the very least enacting some form of church discipline which removes from the individual the joy of fellowship.
The unfortunate situation nowadays is that many people are "lone wolf Christians," or do not attend a church for reasons outside their control, and hence have no primary authority to be disciplined by. Some things to consider:
First, if someone is a lone wolf Christian, then they are in violation of scripture's command to be a part of corporate worship (cf. Heb 10:25), and this suggests some deeper spiritual issues with the individual. I would put forward, if a person continues to willingly forsake fellowship even after it has been offered to them, and they are fully capable of engaging in fellowship, then this may be a sign that they are not truly converted, and in this case (but only in the most extreme scenario), the person may be already considered "as a Gentile and a tax collector."
Second, if the person has a legitimate reason they are not part of a fellowship of believers, then the solution will most likely be on a case-by-case basis, and at the discretion and conviction of those who are exhorting him to repent. Obviously, the ultimate goal would be that the person be brought to repent.