Rav was one of the first generation rabbis of the Amoraim - the Sages responsible for the Gemara. He, together with Shmuel (as well as Rabbi Yochanan in the next generation who was also a student of his) was of the greatest of the Rabbis of this period. He died in the year 246. His name was actually Abba (Arika) - but was given the title "Rav" in Bavel, where he lived - much like Rabbi Yehuda HaNasi, the editor of the Mishnah was given the title "Rebbi" - as a means of respect. Some say that wherever the Gemara says "והאמר מר" - "But didn't the master say" that this is none other than Rav himself.
He ascended to Eretz Yisrael to learn from Rabbi Chiya who was also his uncle and his main teacher! Rabbi Yehuda HaNasi was also his main teacher! It was he who called Rebbi "Rabbeinu HaKadosh", "Our holy Rabbi". When he returned to Bavel, he would send his questions to Rebbi who lived in Eretz Yisrael. He returned later to Eretz Yisrael and supposedly stayed there until the death of Rebbi. He returned to Bavel in the year 218 and neither Shmuel nor the other Chachamim had heard of him! This would change soon as Shmuel would become a good friend of his! Never forget that while the opinions of these great Sages may differ, they were most often the best of friends! One of the greatest levels one can achieve in learning Talmud is to be able to understand the value of each opinion and the greatness which it presents in teaching us the Halacha as we need to follow it, each person, in every generation!
Rav was even greater than Shmuel who humbled himself to Rav. He formed a Yeshiva in Sura while Shmuel formed his Yeshiva in Nehardeah. His Yeshiva remained for some 800 years until the period of the Geonim. He had thousands of students. Just as there are the famous discussions between Ravina and Rav Ashi, so too is the Gemara filled with the discussions (arguments!) between Rav and Shmuel. His students called him "Rabbeinu HaGadol" - our Great Rabbi. In matters of ritual law, the Halacha follows Rav. In matters of money law, the Halacha follows Shmuel.
When coming to Bavel, Rav was extremely poor having practically nothing to eat! He did however become rich! He taught his son Chiya.
He died at an advanced age leaving behind thousands of mourners in Bavel.