The parsha of Balak, which we read last week, concluded with the daughters of Moav seducing the Jews into idol worship. In the midst of that, one of the Jewish princes took a Midianite woman for harlotry. A man named Pinchas zealously took a spear and killed them both in the act, in a miraculous way.
The first aliyah in this week's parsha begins by continuing that story, with G-d praising Pinchas. The Torah identifies the slain man as Zimri a prince of the Jews, and the slain woman as Cozbi, the daughter of a Midianite king. Rashi points out the hatred of the Midianites toward the Jews: that they even offered their king's daughter for prostitution for the sake of causing the Jews to sin. G-d then commands the Jews to harass and smite the Midianites.
At the very end of the previous aliyah, G-d commands that the Jewish people be counted again. Therefore this, the second aliyah, names all the families and their counts. Rashi points out that each family name is expressed as the father's name, with the Hebrew letter 'hay' at the beginning, and a 'yud' at the end. Rashi says that these two letters are representative of G-d's name and are G-d's testimony that the Jews descended from their Jewish fathers and that the Egyptians had no immoral relationships with their mothers.
After the counting of the Jews in the previous aliyah, in this, the third aliyah, G-d commands Moshe to divide up the land of Israel as an inheritance according to the size of each tribe. Although the size of each portion is determined by the size of each tribe, the location of each portion within the land of Israel is determined by lottery. The Torah states here that all men previously counted by Moshe died in the desert, except for Calev and Yehoshua. All women lived because they didn't believe the evil report of the spies.
At the end of the aliyah, the daughters of Tzelafechod approach Moshe and state that their father died in the desert of his own sin (unrelated to the sin of the generation that listened to the spies), and left no sons. Therefore, according to the law of inheritance stated so far, their family would get no portion of the land for an inheritance. Their answer comes in the next aliyah.
As we said in the previous aliyah, the daughters of Tzelafechod brought a claim to Moshe. In the fourth aliyah, G-d answers that they are right, that if a man dies and has no son, his inheritance goes to his daughter. Then this aliyah goes on to explain other laws of inheritance.
The aliyah continues with G-d telling Moshe to go to the top of the mountain to see the land that the Jews are going to. G-d says that after that, Moshe will pass away without entering the land, because of the incident at the waters of Maribah (that was discussed in parsha Chukas). Rashi points out that Moshe immediately puts aside his own interests and asks, for the sake of the community, that G-d appoint a leader to succeed him. So at the end of the aliyah G-d appoints Yehoshua to take over in Moshe's place.
In the fifth aliyah we read the descriptions of the continual daily offering to be offered twice each day, once in the morning, and once in the afternoon. These verses are recited every morning and afternoon before the Shacharis and Mincha.
Also described here are the Sabbath offering and the Rosh Chodesh offering, which are read as part of the Musaf prayer on those days. In all these cases these passages are currently read as part of our prayers as a substitute for the actual services in the Bais Hamikdash, may it be speedily rebuilt in our days, Amen.
In the sixth aliyah we read some laws of the observation of Pesach beginning on the 14th of the month of Nissan and continuing for seven days of eating matzo. Then we read about the observation of Shavous, and Rosh Hashanah, with the blowing of the shofar. After that, the Torah discusses Yom Kippur, where we are to afflict our souls for a full night and day, in repentance. The oral law defines afflicting one's soul as abstaining from eating, washing, anointing, wearing leather shoes, and marital relations.
In the seventh aliyah we read about the holiday of Succos with a different order of offering for each day. On the first day 13 bulls, 12 the second day, and so on, 1 less each day for a total of 70 bulls offered. Rashi says that the 70 bulls refer to the 70 non-Jewish nations, and during the times of the Temple, these offerings of 70 bulls protected the 70 nations from afflictions. Yet those same nations didn't appreciate that fact and destroyed the Temple.
Following the description of Succos, this aliyah discusses the holiday of Shemini Atzereth. This is the last holiday in the yearly chronological sequence of holidays. And Rashi says this is like a father whose children are about to leave. He says to them: please stay one more day with me. And that's what Shemini Atzereth is like: one more day together with G-d at the end of all the other holidays.