The best way to conceive of the conflict in this chapter is to say that it is a conflict between Balaam and God (though you could argue that it is between Balaam and himself). The conflict is resolved by Balaam's decision (though it is somewhat coerced) to obey God.
Balaam is in conflict with God because Balaam is greedy for wealth and prestige. He wants to be able to curse the Israelites so that Balak, King of Moab, would give him wealth and honors. However, God does not want Balaam to curse the Israelites because they are God's people. Balaam keeps arguing with God. Finally, God puts an end to the conflict by sending the angel that prevents Balaam from going to curse the Israelites.
Thus, the conflict of the story is resolved by God sending the angel to convince Balaam (with the help of Balaam's donkey) to obey God.