Many of the lessons of yesterday's passage are carried through to this one, especially the fact that we are saved by faith, not works, but there are also a number of useful lessons to consider here. (obviously, since it is God's word.)
One such lesson is found in verses 15-18. James is discussing the fact that God has seen fit to save the gentiles, and states: 'the words of the prophets are in agreement with this'. This is important. I think too often we forget that the old testament is just as applicable and useful as the new testament. 2 Timothy 3:16 says that all scripture is God breathed, and therefore the old testament is just as good and true and right as the new. Nothing in the new testament has not already been in the old, and God never changes. Of course, through Christ a number of things have been fulfilled (eg: the sacrificial system was fulfilled by him being the sacrifice for our sin once for all), but there are still plenty of lessons which we can learn from it, and we must not neglect it! None of God's words can ever be useless, it is all instructive and crucial to the life of a Christian.
Another lesson from these verses (v 15-18) is that God keeps his promises, and proves his people right. Throughout the old testament, people have been told by God to prophecy certain things, and God has never forgotten his promises through these people. In Genesis he promised a son from eve who would 'crush the serpents head' - defeat Satan (Gen 3). Who did that? Jesus! That was perhaps 4000 years before Jesus was even born! There are plenty of other examples - especially in Isaiah and the Psalms. God keeps his promises, and so we can trust him to keep his promise to save us eternally, to forgive us if we repent and believe and to bring us to his kingdom in heaven forever. What a blessing! As a side note, from this it's obvious that God is in control and always planned to save us by sending Jesus.
Lastly, I wanted to put forth some ideas about verse 20. In this verse we have James saying that the Gentiles should be told not to: 1. Eat food offered to idols, 2. Be sexually immoral or 3. Eat strangled animals or blood. I've done a little research and I'm pretty sure these are the reasons for the commands. 1. Don't eat meat offered to idols is commanded for two reasons: Eating that meat will allow Satan to tempt the Gentile believers to worship the idols also, so to prevent possible temptation and sin, and secondly - eating food sacrificed to idols would be a stumbling block to the jews and would cause then to sin. Paul also deals with this in 1 Corinthians 8. We must by do anything which may offend fellow believers or cause them to sin. 2. Sexual immorality is obviously sinful and condemned throughout scripture, but in the life of Gentiles, it was quite common, and was even a way to worship their idols. This is commanded so that Gentile believers will not fall into this sin without realizing that it is sinful. 3. Blood and strangled animals again seems to be simply because these things will cause offense to the Jews and possibly cause them to sin or else cause disunity in the church. We should be caring of each other and work hard not to offend fellow Christians (except of course with challenging people about there sins which is necessary and often offensive. In general we should seek to be at peace with all men, but never tithe extent that we don't follow clear biblical commands).