"We did not give in to them for a moment, so that the truth of the gospel might remain with you." NIV translation
Paul and his companions did not compromise on the issue of grace. When you add keeping any part of the law, as a requirement for salvation, it is no longer by grace. Then, you get more and more people trying to add another rule (law) and, before you know it, the gospel is distorted into being about what we do instead of what God has done.
"As for those who seemed to be important - whatever they were makes no difference to me; God does not judge by external appearance - those men added nothing to my message." NIV translation
God does not care about the title that the world gives you. Paul tells us that the leaders of the church in Jerusalem did not have anything to add to his message.
"On the contrary, they saw that I had been entrusted with the task of preaching the gospel to the Gentiles, just as Peter had been to the Jews. For God, who was at work in the ministry of Peter as an apostle to the Jews, was also at work in my ministry as an apostle to the Gentiles." NIV translation
The leaders in Jerusalem were able to see that God had given Paul his ministry to the Gentiles. It did not replace any ministry (such as Peter's) but it was also something that God had planned for Paul to do. Some people get upset or jealous over another person doing the ministry that God has laid out for them or even try to do it themselves. When the ministry is from God, it will be obvious for others to see just as it was here with Paul.
"James, Peter and John, those reputed to be pillars, gave me and Barnabas the right hand of fellowship when they recognized the grace given to me. They agreed that we should go to the Gentiles, and they to the Jews." NIV translation
The elders in Jerusalem simply told Paul to do what God had told him and that they would do what God had told them. When God is in control and people see it, then we should all be willing and able to yield to His will. That is what happened here and, if we would all simply trust and listen to God, it could happen with all of the ministries and groups of believers around the world today. What a beautiful sight it would be to see the Body of Christ functioning as He planned it!