Genesis is the bad guy of Crisis Core. He was once a First Class SOLDIER but was experimented on. The experiment was to create the ultimate soldier. The results of this experiment ended up becoming Deep Ground. Following on from this project, Shinra carried on creating another soldier using a similar process, this time using Jenova Cells. The result was Sephiroth. In Crisis Core Genesis escapes Shinra but was hunted down by another SOLDIER called Angeal, who lost his life in the process of sealing Genesis away deep under Midgar. You play as Zack who accompanies Angeal on his mission.
So basicly, Genesis and Sephiroth are similar but are from two diffrent projects. The G Project and the Jenova Project. Deep Ground and the Tsevites are the 'Spawns' of Genesis so to speak, they have no real connection to Sephiroth in general. Angeal is the mentor of Zack and is the orginal owner of the Buster Sword. After Angeal dies to seal Genesis its obviously passed down to Zack.