Brazilian Jujitsu is one of, the if not the premier form of close contact floor fighting used in the world today. It started when a Japanese man, Maeda-san (also known as Count Koma), who was working on forming a colony in Brazil secretly taught a local to perform Jujitsu. The local's last name was Gracie and he passed on the knowledge to his family and a huge family it became. This family has worked together to improve the art of Brazilian Jujitsu into what is sometimes known as the Gracie style of Grappling.
Some of the Gracie family's names are Helio, Carlos, Carlson, Rolls, Rorion, Rickson, Royce, and Renzo among many others. I have compiled a very rough family tree to quickly summarize some of the details. There are numerous non relatives as well who have grown to prominence studying under the Gracies.
Following Carlos the original founder, Helio became the mainstay of the school. In the 1950's & 60's the torch passed to Carlson for about ten years. Carlson preferred to teach more practical moves in contrast to Helio's like of more technical teaching. After that, in the 1970's Rolls and Rickson became the main teachers and fighters. Rolls unfortunately died in a hang gliding accident. But Rickson remains pretty much undefeated to this day.
The torch then passed partially to a new generation in the 1980's, 90's and today with Rorian bringing the Gracies to Hollywood in the form of the UFC challenges and to the world in the form of the Tokyo Pride fights. Royce became famous for doing well in first few UFC titles and for the start of the Pride matches as well.
The Gracies became more even more world renown after their wins and appearances in the various UFC fights and the Pride fights. The losses either push the Gracies to further improve their style, or are sometimes because their opponents are actually students in the Gracie schools or studying the Gracie's techniques. One example of this is the fight in which Helio finally lost to a Japanese fighter named Kimura lead the Gracies to further practice the very move that took him down and named it the "Kimura".
Renzo is one of the latest in a long line of Gracie trainers and teachers. The Gracie's style involves the type of fighting used once one is thrown to the floor and how to incapacitate the other opponent whether they are standing or sitting. Besides defending yourself the objective is to get the opponent into a position in which they can no longer fight back or escape which is often called mounting, but you need to actually mount the opponent.
For more information on the Gracie style of Brazilian Jujitsu, you can read: "The Gracie Way" by Kid Peligro.