When Grizabella was a kitten, she was very beautiful and loved - like Victoria and Jemima. But soon she felt the need to break away from the tribe and explore the rest of the world. This really isn't allowed in the Jellicle Tribe. Jellicle's aren't suposted to socialize with other cats outside the tribe, so she abandonned the tribe.
She led a tough life, with no help from the Jellicles, and saw more of the world then she wanted to. Mostly the bad side of it - she became a cat prostitute.
She returns to the tribe on the night of the Jellicle Ball, but is shunned by the Jellicles. They don't want to have anything to do with her and won't forgive her for what she has done. The female cats find her very offensive. Jellylorum and Jennyanydots protect the kittens, and keep them away from Grizabella. They want to preserve the kitten's innocence. Munkustrap is on guard as well, when Grizabella is around. Although he doesn't see her as a dangerous threat.
It is not until after Grizabella sings "Memory" that the cats finally understand what she has gone through. Victoria is the first to touch her, and she is accepted and forgiven by the all the members of the tribe. And Old Deuteronomy chooses her to go to the Heaviside lair.
Grizabella is either Gus or Old Deuteronomy's mate. some say she's the mother of Munkustrap, Rum-Tum-Tugger, and/or Macavity.