Many people believe that Judaism started with Adam and Eve. Notice that there son's Cane and Able made sacrifices to the Lord on an altar. But many more people trace it back to Abraham.
It began officially in 1332 B.C.E., when the Jews were given the Torah by G-d at Mt. Sinai. However, the ball already began rolling roughly 450 years before when G-d promised Abraham that his children would one day become G-d's nation. This promise was then carried out through Abraham's grandson Jacob's children; they stuck together as a nation and eventually ended up becoming the Jews.
It is possible that they began as a wandering warrior tribe who were allowed to settle in the northern area of Egypt to help protect that border. There are Egyptian records of a tribe called "Habara". Egypt at one point required that the Habara leave their northern territory and live in the cities to help build the pyramid for Ramses. Egypt was worried about the Habara's increasing numbers and worried about their loyalty. The Habara didn't appreciate living in the cities because they were used to governing themselves in their own territory.
Around 50 years before this, the Pharaoh Akhnaton broke with Egyptian tradition and established a new religion with one god - the sun. As soon as he died, the Egyptian people returned to their old religion and tried to eliminate the records of Akhnaton. His attempt to create a monotheistic religion was a disaster. Yet, some people continued to follow it.
It is possible that the Habara were influenced by this religion. At the time of the Exodus, not only the Jews (Habara?) left Egypt but many different ethnic groups left alongwith them. Some of them remained with the Jews and some left. Moses needed to unite these diverse people and that is why he developed the laws of the Torah (including the "ten commandments").
And so what we now know as "Jews" probably originated as a large group of diverse people united by a set of common laws and influenced by a short lived monotheistic Egyptian religion.