The tribes are named after the sons of Jacob. The patriarch Jacob had his name changed to Israel after wrestling with and prevailing over an angel (see Genesis 32).
Levi, though one of Israel's son, is usually omitted from Biblical passages listing the tribes. The descendants of Levi were consecrated as priests to God and were responsible for the service of the tabernacle in the wilderness and later with the temple in Jerusalem. Instead of receiving land as an inheritance in Palestine, the Levites received tithes and offerings from the people (Numbers 18, Deuteronomy 18, etc.).
Although Joseph was also one of Jacob's sons, he too is usually omitted from lists of the tribes. This is because his two sons, Ephraim and Manasseh, were adopted by Jacob in his stead (Genesis 48).
How did the Israelites end up in Egypt?
Jacob's favorite wife Rachel had a child named Joseph. He became his father's favorite son because he was born when Jacob was old (Genesis 37:3). Joseph was so loved by his father that he received from him a coat of many colors (Genesis 37). Joseph's brothers, envious of his special treatment by their father, wanted to kill him. Reuben, Jacob's firstborn son, kept Joseph from being murdered only to have him sold to slave traders on their way to Egypt. Joseph is soon bought by one of Pharaoh's officers.
In Egypt God is with Joseph and blesses him. After a series of events and trials (see Genesis 39-41) he is made ruler over Egypt:
" . . .'You shall be over my house, and all my people shall be ruled according to your word; only in regard to the throne will I be greater than you.' And Pharaoh said to Joseph, 'See, I have set you over all the land of Egypt.' " (Genesis 41:40-41, NKJV)
After Egypt experiences seven years of bountiful harvests (where Joseph stores extra food) a famine overtakes not only the country but also the land of Canaan where Jacob and his entire family are living.
In 1685 B.C. Jacob and his entire household (which includes Joseph's eleven brothers) move to Egypt's land of Goshen in order to survive the famine. Jacob (Israel) lives 17 years in Egypt and dies at the age of 147. Joseph lives to the age of 110.
When did they LEAVE Egypt?
Sometime after the death of Joseph a new king reigned over Egypt. The new Pharaoh, out of fear the growing and increasingly prosperous children of Israel might turn against him, has them made into slaves.
After more than a century of slavery God sends Moses and Aaron to free the children of Israel out of Egyptian bondage. It is only after a series of ten plagues are brought on Pharaoh and Egypt that they are finally released from slavery. They leave Egypt in 1445 B.C. during the reign of Pharaoh Thutmose III.
When did they receive their inheritance?
After the Exodus, instead of entering the land of Canaan, the tribes are made to wander the wilderness for forty years because of their sins (Deuteronomy 8:2). After this period Joshua leads the people into the promised land in 1405 B.C. Seven years later, in 1398, the land of Canaan is divided by lot among the children of Israel (Joshua 14).