Last week's parsha, Vayishlach, had Yakov Avinu returning to his father's house after 22 years away.
This parsha opens in this first aliyah with Yakov dwelling (the meaning of the opening words of the parsha: "Vayeishev Yakov") where his father had lived, in Chevron. This aliyah begins the interesting story of Yosef, Yakov's firstborn son from his wife Rachel. Here we find out that Yakov favored Yosef, and made him a fancy coat, and that Yosef reported bad things about his brothers to their father. Finally, Yosef had dreams implying that he would eventually rule over his brothers. All this added up to his brothers hating him and as we'll see in a later aliyah, they'll plot to kill him.
In the second aliyah Yosef's brothers go off to tend to their father's flock in Shechem. Yakov sends Yosef to see how the brothers and the flock are doing. They see him approaching and plot to kill him, but Reuvain suggests that they don't kill him, and instead throw him into a pit. Reuvain's plan was to secretly rescue Yosef from the pit, but in the end he never really gets a chance to rescue him.
In the third aliyah, when Yosef arrives where his brothers are, they strip him of his fancy coat and cast him into a pit. They subsequently sell him to a caravan traveling to Egypt where he is sold to Potifar, the captain of Pharaoh's guard. Meanwhile, the brothers dip Yosef's coat into blood, and show it to their father Yakov as evidence of Yosef's being devoured by a wild beast. Yakov then begins what Rashi points out will be 22 years of mourning while Yosef is missing. Rashi further says that these 22 years where Yosef doesn't honor his father by letting him know he is alive correspond to the 22 years that Yakov himself didn't communicate with Yitzchak his own father while Yakov was in Lavan's house.
In the fourth aliyah the story of Yosef is interrupted by the episode of Yehuda and Tamar. Yehuda is one of Yosef's brothers, in fact the one who suggested selling him. After their father Yakov is grieved over the loss of Yosef, Yehuda goes off and marries and has three sons, Er, Onan, and Shelah. He found a wife named Tamar for his first son Er, but Er spilled his seed and died as a result.
Yehuda had his second son fulfill the mitzvah of Yirbum, of marrying his childless deceased brother's wife, but he also, spilled his seed and died. Yehuda hesitated to give his third son to Tamar, so she returned to her father's house as a widow.
Eventually, Yehuda's own wife passes away and he goes on a business trip. Tamar dresses like a prostitute and sits by the side of the road. Yehuda doesn't recognize her and visits her and leaves his signet ring, and she becomes pregnant. A few months later, when her pregnancy is evident, Yehuda orders her to be burned. While she is being taken out to be burned, she produces the signet ring and says the owner of this is the father. She is very careful not to embarrass Yehuda in public (she would rather be burnt if he didn't admit he was the father). From this we learn to what lengths we must go to avoid embarrassing someone.
In any case, the child from this union begins a long line that leads to King David, then on to Moshiach, indicating that this entire story was a Divinely orchestrated plan in fulfillment of the purpose of creation: the Messianic era, may it be fulfilled speedily in our days.
In the fifth aliyah the Torah returns to the story of Yosef. Yosef has been brought down to Egypt and sold to Potifar, the captain of Pharaoh's guard. The Torah says that G-d is with Yosef, therefore Yosef and even Potifar do quite well during this time.
In the sixth aliyah, after a while of Yosef's success in Potifar's house, Potifar's wife becomes very attracted to Yosef because he is very handsome. She makes advances toward him, but he steadfastly refuses her. One time during one such advance, he runs away, but she holds on to his coat. She uses the coat as evidence saying that it was he who made the advances, and Potifar has Yosef put in prison. But G-d is with Yosef still, and he does very well in prison, and he is put in charge of all the prisoners.
In the seventh aliyah, two of Pharaoh's staff are put in prison: his butler and baker. They both have dreams, which sadden them, and Yosef interprets them. The butler dreams about grape clusters from which he squeezes wine out and gives it to Pharaoh. Yosef's interpretation is that the butler would be restored to Pharaoh's service, which turns out to be true. Yosef asks the butler to mention his plight to Pharaoh, but the butler forgets to do it.
The baker dreams about baskets of baked goods that a bird ate. Yosef's interpretation is that the butler would be hung, which also turns out to be true.