Until the previous parsha, Shlach, the Jewish nation in the desert thought they were but three days away from entering Israel, their Promised Land. However, in that previous parsha, the Jewish people sent spies ahead to spy out the land. The spies brought back an evil report about the land, and the Jewish people believed them. As a result, the Jewish people are punished by having to wander 40 years in the desert before entering the land.
In the first aliyah we read of a man named Korach in the Kehos family within the tribe of Levy. He starts a rebellion against Moshe along with 250 men of renown from the congregation. They claim that Moshe and Aharon are no holier than the rest of the congregation. Therefore they want to know why do Moshe and Aharon act in a special capacity, above the rest of the congregation? Moshe answers that in the morning they will hear G-d's decision.
In the second aliyah Moshe tells Korach and his 250 followers in the rebellion to prepare a firepan and incense and present it before G-d. In addition, Korach goes around to all the Jewish people and incites them also against Moshe and Aharon. Everyone comes to the door of the tent of meeting and the Glory of G-d appears to them there. What happens then is in the next aliyah.
In the third aliyah G-d tells Moshe and Aharon to separate themselves from this community so G-d can destroy them in an instant. Moshe and Aharon fall on their faces and pray to G-d to not destroy everyone for the sake of the few who sin. Actually, according to Rashi, they pointed out that G-d is not like a mortal king who doesn't know a person's thoughts. Therefore a mortal king is likely to punish many because of a few, because the mortal king doesn't know who the guilty few are. But G-d knows each and every thought of each and every creature; therefore, G-d can distinguish the good from the bad.
G-d then made the earth opens up and swallows Korach and the ringleaders of the rebellion and their families, and fire consumes the rest of the 250 rebels.
The next day, the whole community starts complaining to Moshe and Aharon that they killed G-d's people! Once again the cloud of Glory appeared and ... (to be continued in the next aliyah)
In the fourth aliyah, just like the previous aliyah, G-d tells Moshe and Aharon to separate themselves from the community so G-d can destroy them in an instant. And again Moshe and Aharon fall on their faces. But this time Moshe tells Aharon to quickly take a firepan and incense and go into the midst of the congregation because Moshe says that a plague is beginning while they speak. Aharon runs to do this and stops the plague. Actually, Rashi says that once when Moshe visited heaven, an angel told him this technique of using incense to stop a plague. Rashi also says that when Aharon went into the midst of the people he grabbed and physically restrained the angel making the plague. The angel complained and asked that he be permitted to do his job. Aharon told him that Moshe commanded him to stop the angel. The angel said that he was sent by G-d and Aharon was only sent by Moshe. Aharon replied that Moshe only speaks the will of G-d and he pointed out that G-d and Moshe are standing together, and that the angel can go over to them and ask about it.
The fifth aliyah describes the test of the staffs. In order to prove which tribe is divinely chosen to do the priestly service, G-d tells Moshe to take a staff from each tribe with the name of that tribe's prince on it, and the name of Aharon on the staff of the tribe of Levi. These staffs are to be placed in the tent of meeting. This is done, and in the morning it is found that Aharon's staff sprouted leaves, almond blossoms and even almonds! This proves that Aharon is G-d's choice of Kohane Gadol.
The sixth aliyah begins with the Jews expressing to Moshe their fear of dying by mistakenly coming too close to the Mishkan. G-d then commands Aharon and his sons and the Levites to guard against any such mishap by carefully guarding the Mishkan from any common people approaching too close. After that G-d enumerates what Rashi calls the 24 gifts of the priesthood. These include the privilege of eating of the sacrifices the Jewish people bring as well as receiving the five shekels that Jews use to redeem their first-born sons in a Pidyon-haben ceremony. The aliyah ends with G-d stating that the Kohanim won't receive a portion of the land of Israel like the other Jews, instead G-d says that He is their portion.
In the seventh aliyah, G-d says that the Leviim, like the Kohanim, won't receive a share of the land of Israel. Instead they will get the tithes of the other Jews. Furthermore, of this tithe that the Leviim receive from the Jewish people, the Leviim must, in turn, separate a tithe from it and give it to the Kohanim. The balance the Leviim keep as a wage for their service.