Verse 8: (For he that wrought effectually in Peter to the apostleship of the circumcision, the same was mighty in me toward the Gentiles:) In other words, just as God was working mightily through Peter with the kingdom gospel to the Jews, so too God was working through Paul with the grace gospel to the Gentiles.
I think too that given the fact that Paul's apostleship was called into question many times because his calling was as of one born of due time (1Corinthians 15:8), he wanted to make it clear that he was being used just as effectively as Peter. He is also magnifying his apostleship to the Gentiles (Romans 11:13). Of course, any challenge that Paul's apostleship had come from the Jews.
Verse 9: And when James, Cephas, and John, who seemed to be pillars, perceived the grace that was given unto me, they gave to me and Barnabas the right hands of fellowship; that we should go unto the heathen, and they unto the circumcision. Interesting that Paul decides to call Peter, Cephas, in this verse. Some would argue that he is speaking of someone else, but who could that be in the company of James and John?
Also, interestingly, this is the first and only time John is mentioned in Paul's writings. His point is that these three recognized a grace that was given to Paul and extended the right hands of fellowship; and that fellowship meant that they were all working toward the same end, but to different groups; they unto the circumcision, and Paul unto the heathen.
Also, note the word perceived. In the original it means to know experientially (ginosko). In other words, Peter, James, and John, experientially knew that God was doing something different through Paul that he was not doing through them. They recognized the two different programs that we can seem to grasp today.
Verse 10: Only they would that we should remember the poor; the same which I also was forward to do. This seems to be referring to one of the outcomes of the Jerusalem Council. However, Luke never recorded it in Acts 15:28-29.
Barnes says of this, "That is, as I suppose, the poor Christians in Judea. It can hardly be supposed that it would be necessary to make this an express stipulation in regard to the converts from among the Gentiles, and it would not have been very pertinent to the case before them to have done so." It makes sense that the Jerusalem church was already suffering quite a bit since selling all of their possessions. This would also explain why Paul made collections for them (1Corinthians 16:1-4). In the end, just because it was not written in Acts 15 does not mean that it was not discussed.
Verse 11-12: But when Peter was come to Antioch, I withstood him to the face, because he was to be blamed. For before that certain came from James, he did eat with the Gentiles: but when they were come, he withdrew and separated himself, fearing them which were of the circumcision. Whenever this event happened, it was obviously after the counsel in Acts 15 because Peter was still functioning under the Law in the presence of them which were of the circumcision, but not so when they were not around.
Those he refers to that certain came from James would have been part of the little flock or kingdom believers. At issue was Peter's duplicity, i.e., hypocrisy.
So many have this notion that from Acts 2 following, the Jews did not keep the Law. The Scripture simply does not teach that. Acts 21:20-21 says, And when they heard it, they glorified the Lord, and said unto him, Thou seest, brother, how many thousands of Jews there are which believe; and they are all zealous of the law: And they are informed of thee, that thou teachest all the Jews which are among the Gentiles to forsake Moses, saying that they ought not to circumcise their children, neither to walk after the customs. These verses make it clear that they were still zealous of the law.
Another question that I have is why was Peter in Antioch? I believe that he was wanting to see firsthand what God was doing among the Gentiles through Paul's ministry. No doubt, the fact that Paul rebuked Peter, and he received it, speaks to the newfound authority that Paul had and the Twelve recognized.