These notes are based on the NASB text.
5:1 It is actually [and factually] reported that there is immorality among you, and immorality of such a kind as does not exist even among the Gentiles, that someone has [co-habits with] his father’s wife [step-mother].
The report that Paul received from Chloe’s people (1:11) contained more than the disappointing news of factions, it also contained the shocking news of incest. A believer was actually involved in an immoral affair with “his father’s wife” (step-mother). It is safe to assume that the woman was not a believer but a pagan since Paul did not mention the church disciplining the woman. Such a sin did “not exist even among the Gentiles.” Even the Gentiles did not tolerate such a thing. Roman law forbade such relationships. “The ‘Institutes of Gaius’ lists persons related to a man whom he is not free to marry” (Broadman Bible Commentary, Volume 10).
Here we see a believer living below, rather than above, the world’s standards. Such activity undoubtedly damages the witness of believers, both individually and corporately.
5:2 And you have become arrogant, and have not mourned instead, in order that the one who had done this deed [obviously a carnal Christian] might be removed from your midst.
The only thing worse than the man’s sin was the church’s tolerance of his sin.
How had the Corinthians become “arrogant”? By taking pride in their broad-minded tolerance of this man’s sin. Such tolerance communicated to an unbelieving world that even a fornicator engaged in blatant sin could be a member of the church in good standing. Such tolerance misrepresented and discredited the cause of Christ in the community. Such tolerance erased the line of demarcation between believers and unbelievers.
What should the church have done? The church should have “mourned instead.” The word “mourn” means to grieve as for the dead, a deep and painful kind of sorrow. Further, the guilty individual should have been removed from the fellowship.
5:3 For I, on my part, though absent in body but present in spirit, have already judged him [the only response to such a sin that is contrary to God’s Word] who has so committed this, as though I were present.
5:4 In the name of [by the authority of] our Lord Jesus, when you are assembled [this was such a serious matter that Paul did not want for the Corinthians to wait until he arrived to deal with it], and I with you in spirit, with the power of our Lord Jesus,
5:5 I have decided [Paul assumed the man’s lack of repentance] to deliver [a decision that must be made by the entire church] such a one to Satan [to exclude from the church and send back to the world which lies in the power of the evil one – see 1 John 5:19] for the destruction of his flesh [either physical death or a purging of a sin-centered orientation], that his spirit may be saved in the day of the Lord Jesus.
The church was to approach this matter in keeping with the instruction of Jesus in Matthew 18:15-20. If the individual failed to repent, he was to be dismissed from the fellowship. This was a public sin and was therefore to be judged publicly. Such action was to be done with a redemptive, not vindictive, view, “that his spirit may be saved in the day of the Lord Jesus.” The hope was that the man would be shocked into repentance once outside the beneficent protection and care of Christ.
Such disciplinary action is  for the welfare of the individual,  for the protection of the church from the influence of such individuals, and  for the witness of the church in the world.
5:6 Your boasting is not good [fitting]. Do you not know that a little leaven leavens the whole lump of dough?
Once again we see the attitude of the church toward the situation under consideration. The church was “boasting,” perhaps of its broad-minded tolerance of the sin or of their forgiving spirit. Paul, however, warned the church that this attitude was “not good” (or not fitting) because it endangered both the witness and future of the church. Such tolerance or indifference toward this sin had the potential of ruining the moral fiber of the entire fellowship. Others might be encouraged to do the same. Paul illustrated his point with the reference to “leaven” which is a picture of sin/evil in the Scripture. Just as a small amount of “leaven” (yeast) gradually permeates a whole lump of dough, so one sin has the potential to spread like a cancer through a congregation.
5:7 Clean out [immediately and without hesitation] the old leaven [the things pertaining to the old life], that you may be a new lump [a church with a proper attitude toward sin and morally pure], just as you are in fact unleavened [become what you already are, begin acting like new creatures]. For Christ our Passover also has been sacrificed.
Alluding to the Old Testament Passover, Paul instructed the Corinthians to remove evil/sin from their midst even as the Jewish families were instructed to search for and remove any leaven from their homes prior to celebrating the Passover (see Exodus 12:15).
5:8 Let us therefore celebrate the feast, not with old leaven, nor with the leaven of malice and wickedness, but with the unleavened bread of sincerity and truth.
Paul urged the Corinthian believers to live their Christian lives without “the leaven of malice and wickedness.” They were to remove all “old leaven” from their personal lives. Only then could their lives be like unleavened bread. Only then would their lives be characterized by “sincerity and truth.”
5:9 I wrote you in my letter [an earlier letter that did not become a part of the New Testament] not to associate [mix with] with immoral people;
Paul warned the Corinthians not to fellowship with immoral people, like the man living with his father’s wife. This for at least two reasons:  because “Bad company corrupts good morals” (see 1 Corinthians 15:33), and  because the witness of the church in the community would be compromised (see 1 Corinthians 15:34).
5:10 I did not at all mean with the immoral people of this world, or with the covetous or swindlers, or with idolaters; for then you would have to go out of the world.
It is one thing to fellowship with immoral people and quite another to go fishing for them! Apparently some who read Paul’s previous letter thought that Paul had suggested that they have nothing to do with any immoral person. This was not possible, “for then you would have to go out of the world.” There is a difference between isolation and separation. We can certainly fish for immoral people without fellowshipping (getting mixed up with them in an intimate way) with them.
5:11 But actually, I wrote to you not to associate [get mixed up with in an intimate way] with any so-called brother [a church member or a professing Christian] if he should be an immoral person, or covetous, or an idolater, or a reviler, or a drunkard, or a swindler – not even to eat with such a one [either a reference to the Lord’s Supper or a fellowship meal that might be seen as condoning the sin of an individual – be careful lest social acceptance imply spiritual acceptance].
Paul was not interested in getting the church out of the world but rather getting the world out of the church.
5:12 For what have I to do with judging outsiders? Do you not judge those who are within the church?
5:13 But those who are outside, God judges [He exercises judgment on the whole world]. Remove the wicked man from among yourselves [some believe that 2 Corinthians 2:1-11 refers to the immoral man in question…if so, then the disciplinary action of the church brought the man to repentance].
As Christ’s representatives, believers should live above the world’s standards.
The immoral man of 1 Corinthians 5:1 was living his life below the standards of the world. Such behavior is ruinous to the witness of both believers and the church. As Christ’s representatives in the world, we are to live as new creatures [1 Corinthians 5:17]. We must no longer embrace the old things but cast them aside. We must walk in newness of life [Romans 6:4] and in a manner worthy of our calling [Ephesians 4:1].
The church must exercise discipline when necessary.
The church has a responsibility to exercise discipline in keeping with the guidelines of Matthew 18:15-20. Church discipline must be exercised with a redemptive spirit. Failure to exercise church discipline when clearly called for may cause irreparable damage to the sinning individual and to the fellowship of believers.
Corrupting influences must be removed.
Corrupting influences must be removed from the both the personal lives of believers and the church at large. Each individual has a responsibility to allow God to search his life and remove that which has the potential to harm and damage and destroy [see Psalm 139:23-24].
Tolerance of sin damages the witness of individuals and the church.
Tolerance of sin on the part of Christians and the church obscures the line of demarcation between believers and unbelievers. Unbelievers observing such tolerance/compromise can easily conclude that Christ is incapable of effecting change in the lives of believers.