In last week's parsha, Miketz, Yakov Avinu and his entire family, the progenitors of the twelve tribes, descended into Egypt.
The first aliyah begins with the words: "And Yakov lived 17 years in Egypt." Yosef, his son, left his father Yakov after Yakov fed him 17 years. Now, mida-kneged-mida (measure for measure), Yosef feeds his father Yakov 17 years because his father is now living under Yosef's care in Egypt. Yakov asks Yosef, and Yosef promises, to bury Yakov in the Cave of Machpella. Later, Yosef hears that his father is sick. He goes to visit him (the Midrash says that visiting the sick provides /60 of a cure) and to have his two sons blessed.
In the second aliyah Yakov says he didn't think he'd ever see his son Yosef again, and now he's even seeing Yosef's children. Yakov blesses them "to grow into a multitude in the midst of the earth" (multitude - "vyidgoo" like "dag" - fish, i.e. no evil eye shall ever harm them).
In the third aliyah when Yakov blesses Yosef's sons Ephraim and Menashe, he places his right hand on Ephraim, the younger son, and his left hand on Menashe, the older son. Yosef is concerned about Yakov's right hand on Ephraim, so he tries to remove it, but Yakov affirmed his action and said the younger will be greater. Yakov blessed them: "By you Israel shall be blessed: G-d make you as Ephraim and Menashe", which is how we bless our children on erev Yom Kippur. Yakov said G-d would be with the Jewish People in exile.
In the fourth aliyah Yakov called all his sons to tell them when Moshiach will come, but his prophecy is turned to blessings, instead.
Each tribe is blessed, as follows:
Reuven's blessing: as first-born he was destined for priesthood and kingship but lost them by his hasty actions regarding his father's couch.
Shimon and Levi's blessing: Conspiracy (Zimri from Shimon, who will bring the Midianite woman into the camp for harlotry, and Korach and his rebellion, from Levi).
Yehuda's blessing: Monarchy, Moshiach, and abundance of wine and milk in his portion.
Zevulun's blessing: Supporter of Torah (specifically, the tribe of Zevulun engages in sea trade to support the tribe of Yissachar in their Torah study).
Yissachar's blessing: They were active in Torah study, they bore the yoke of Torah, and from Yissachar issued 200 heads of the Sanhedrin (Rashi).
Dan's blessing: Shimshon (Samson) - a judge over Israel who will avenge the Philistines.
The blessings of the tribes continue in the fifth aliyah:
Gad's blessing: Strength. According to Rashi, this means that Gad, whose portion will be on the opposite side of the Jordan from Israel, will cross over the Jordan to conquer Israel with their brethren, and then return to their portion without even a single man missing.
Asher's blessing: an abundance of olive oil in their portion.
Naftali's blessing: Beautiful fruit, beautiful words.
Yosef's blessing: Charm, righteousness, Stone of Israel, with the evil eye having no power over his sons.
In the sixth aliyah:
Binyamin's blessing: Shaul fighting against Moav and Edom, and Mordechai and Esther in the story of Purim.
Yakov says that his passing away is near, and requests that he be buried in the cave of Machpella, which Avrohom bought in parshas Chaya Sara. After that, Yakov passes away. The Egyptians weep for him 70 days because when Yakov arrived, blessing arrived with him (i. e. the famine ceased [Rashi]). Then a very great funeral procession consisting of all the elders of Egypt, besides Yakov's own family, went up to bury Yakov. Yakov's sons carried the coffin in the same arrangement that their tribes will later journey and camp in the desert.
The seventh aliyah describes the later years of Yosef's life in Egypt. Yosef lived for 110 years. He promises his brothers that G-d will eventually bring them out of this land, and he makes them promise to carry his bones with them, and bury him in Israel (he's buried in Shechem). Yoseph Hatzadik then passes away in Egypt.