These verses occur after the first missionary journey of Paul and Barnabas. They have returned to the church in Antioch, and some Jewish believers come along and say that the Gentiles must be circumcised to be saved. As a result they journey to Jerusalem to ask the apostles. There are three lessons I would like to draw out of this passage.
Firstly, we have the word 'brothers' which is used to describe fellow believers (see verses 1,3 and 7). This is used throughout the new testament, but I want to mention that we, Christians, are members of the same family. If you have genuinely acknowledged your sin and repented of it, and if you have faith in Jesus Christ as Saviour and Lord then you, like me, have been adopted into God's family. Since believers are a family, we should act like it. We should help each other, care for each other, have genuine friendships which we put effort into and we should study God's word together. We should also be completely obedient to God, our father.
Secondly I want to draw out the main point of the passage: faith verses works. The Jewish believers claim in verse 1 that "unless you are circumcised according to the custom of Moses, you cannot be saved". This is a problem, we have some people saying that works are necessary to be saved. Instead of being saved by faith alone, they are claiming that we need to do something to be saved. This cannot be true when we consider the following facts: firstly, we are completely sinful, and we all deserve hell. In Romans 3:23 we learn that all have sinned and in Romans 6:23 we learn that the wages of sin is death. We are in an utterly hopeless situation. Secondly, all that we do is like filthy rags in Gods sight (Isaiah 64:6), so nothing that we can do can improve on our salvation or even please God unless we have been cleansed of our sinfulness. So how does this apply to our lives? Well, we cannot earn salvation. We cannot do anything to save ourselves, and the reverse is also true: nothing we do is to big that God can't forgive us for it. What a comforts that is! If we confess our sins, he is faithful and just and will forgive us our sins and purify us from all unrighteousness. (1 John 1:9). We cannot do anything so bad that God is not willing or able to forgive us for. So don't let Satan chew you up with false guilt - repent and stop sinning! God will forgive!! Along with this point is the fact that genuine faith leads to good deeds. If you are a Christian, then act like it! We are not saved by our deeds, but we are saved to do good deeds - for Gods glory (see Ephesians 2:8-10, 1 Corinthians 10:31 and James 2)
Lastly I want to point out that all Christians are equally saved. Denomination does not matter to God, and we should put next to no emphasis on it. Verse 4 makes that clear: when the believers come to jerusalem they are 'welcomes by the church and the apostles and the elders'. Likewise in verse 9, God made no distinction between the Jewish believers and the Gentiles. If a person believes the basic truths of the gospel, is seeking to learn and grow and is showing the fruits of salvation then we can assume that they are saved and therefore our equals. Likewise if a church holds fast to the truth, teaches from the bible, then regardless of denomination we should regard it as a fellow church. Sure, we will disagree in areas, and we should discuss these things, but they should not get in the way of fellowship with one another or give room for Satan to get in (unless of course they are crucial truths of the bible which cannot be disputed).
I hope that this has been helpful to you and that God uses it to glorify himself.