While people of all races survived the Great War, the term "race" in Fallout world refers mostly to different kinds of human mutants, such as ghouls and super mutants, as well as humans themselves.
Aside from the obvious mutant-ghoul-human and cultural bigotry (tribal's vs. townsfolk, Vault City vs. everyone else), racism and sexism is alive and well. A person's skin color, status, and career are always good excuses for fear and hate to build upon, although communities like NCR and Modoc would have less of a problem with this. The Great War may have brought some communities together, but it also made isolated communities as well. Many races from previous games have made an appearance again in Fallout 3, with introductions to new species like the backwater swampfolk, dead failed FEV subjects, trogs and abomination.
Only Fallout Tactics and Fallout: Brotherhood of Steel allows the player to play another thing than human race. Fallout Tactics, although only in multi-player mode, lists dogs and brahmins among playable races.