In this chapter, Balaam issues his first two prophecies concerning Israel.
After the long, long introduction of the last chapter, this chapter is now relatively straightforward. Having departed for the "high places", i.e. the religious centers of Moab, Balaam is now ready to begin the process of cursing Israel. They construct seven altars (seven being the number of completion or fullness) and make some offerings. It is possible the animal entrails were used for divination in this case, although the text does not directly mention it.
The contents of the prophecies also does not seem noteworthy to me. They are both written in the standard poetic form and it's basically just the LORD lavishing praise on Israel. As I said before, this is probably intended to encourage Israel for the coming battles as they invade the promised land.
What's more interesting to me is the dynamic between Balaam, Balak and the LORD. In particular, Balak and the LORD seem to be positioned opposite of one another, with Balak seeking to curse Israel while the LORD overrides him by putting words into the mouth of Balaam, the prophet. Balaam is stuck in the middle between his desire for money and his insistence that he must speak the words that "the LORD puts in my mouth". In the end, the LORD appears to win out, possibly due to the threatening angel of the last chapter. What's interesting about this is that Balaam is one of the few, maybe the only, true prophet who is not an Israelite. There are lots of false prophets who are not Israelite, but here we can see that the LORD is indeed putting words into Balaam's mouth and I can't think of any other time something like this happens.
In response to these blessings, we see Balak's rising anger at his failed attempts to curse Israel. Balaam appears helpless in response, repeatedly citing that he can only speak what the LORD gives him. We can immediately see that this answer does not assuage Balak, who is himself caught between the hapless prophet and his own immense fear of the Israelites. They try moving to different locations to see if they could possibly curse them somewhere else. The first time they move, in v. 13, it seems the intent was to see only "the extreme end of them and will not see all of them", because perhaps it's easier to curse a few people than to curse many people. Repeating the ritual with the seven altars, this approach clearly doesn't work as Balaam's second blessing is longer and richer than the first.
Balaam gets even more angry, and this time tries to relocate to the "top of Peor". This is probably one of their most sacred high places, because in Num 25 we are told there is a "Baal of Peor". Also, the top of Peor "overlooks the wasteland", so this is a dominating vista. Balak probably expects that relocating here would give him access to more spiritual power as this is probably one of their most sacred shrines. To see the results of this third, final attempt, we will have to read the next chapter, but I'll give you a hint: it doesn't work.