This week's parsha is parsha Balak, about the non-Jewish prophet Bilam.
The first aliyah begins where last week's parsha Chukas left off, where the Jews have just conquered two kingdoms and acquired their land. Now Balak, who was the king of Moav, becomes aware of these victories of the Jews, which mean danger for him and his country. So he sent messengers to Bilam, who was known as a famous non-Jewish prophet, and asked him to curse the Jews. Bilam discusses the matter with G-d and G-d tells him not to curse the Jews.
In the second aliyah Bilam sends word with Balak's messengers that G-d doesn't give him permission to go with them. Actually, Rashi says that Bilam hinted that he would go with messengers who were more honorable than those sent the first time. Balak then sends more honorable messengers to Bilam and promises him fancy rewards. Bilam discusses the matter again with G-d, and now G-d says that he may go, but only to speak such words that G-d tells him to.
In the third aliyah Bilam saddles his donkey and goes with the Moavite dignitaries. However, G-d places an angel with a drawn sword in Bilam's way. Rashi says that animals have better sight than humans, so the donkey could see the angel while Bilam could not. The Torah describes the trouble Bilam has trying to get the donkey to go forward. G-d finally gives Bilam the ability to see the angel. Bilam sees him and becomes afraid and offers to go back. But the angel says he can go on, but only to say what the G-d wants him to say.
In the fourth aliyah Bilam goes with Balak and they build seven altars and make sacrifices to G-d. G-d puts a message in Bilam's mouth and he blesses the Jews saying that he cannot curse a people that G-d doesn't curse, that they are a nation dwelling in peace, and not counting itself among the nations. He ends by wishing that his end be like theirs. Balak, who hired him to curse the Jews, complains, and Bilam tells him that he can only say what G-d tells him to say.
In the fifth aliyah, since Bilam didn't curse the Jews yet, Balak now takes Bilam to another place, namely the top of Pisgah, and Balak hopes that Bilam would be able to curse the Jews from there. Rashi says that Balak knew that this was an unfavorable place for the Jews because this was where Moshe Rabeinu would eventually pass away. They build seven altars and make offerings again, then G-d puts further words in Bilam's mouth. According to Rashi, Bilam says that G-d has already sworn to bring the Jews to Israel, so they cannot be stopped. He says that there is no iniquity or perverseness in the Jews and the friendship of the King is with them.
In the sixth aliyah, since Bilam didn't curse the Jews in the last aliyah, Balak now again takes him to another place, the top of Peor, and Balak hopes that Bilam can curse the Jews from there. Again Rashi says that Balak knew this to be an unfavorable place for the Jews. Now for a third time they build seven altars and make offerings. Then Bilam begins, and blesses Israel a third time, this time with the famous blessing of "How goodly are your tents, oh Yakov" interpreted by Rashi to mean that he saw their tents arranged not opposite one another for the sake of tzniousness (modesty).
In the seventh aliyah Bilam then begins a prophecy about the coming of Moshiach and what he will do to the nations in the final days including the destruction of Amalak. After this, Bilam and Balak go home.
Following Bilam's unsuccessful attempt to curse the Jewish nation, Moavite young ladies come out and seduce Jewish men to be involved with Baal Peor, one of the Moavite gods. Rashi says that they did this on the advice of Bilam. G-d becomes angry and commands Moshe to hang the guilty people. Then, a Jewish man named Zimri publicly takes a Midianite woman to commit harlotry with her, and Pinchas, Aharon's grandson, takes a spear and kills them both in a miraculous way.