While people of all races survived the Great War, the term "race" in the 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 (tribals 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 has isolated others as well. Many races from previous games have made an appearance in Fallout 3, while it also introduces new ones like the backwater swampfolk, dead failed FEV subjects, trogs and abominations.
Only Fallout Tactics and Fallout: Brotherhood of Steel allow the player to play as anything other than the human race. Fallout Tactics, although only in multi-player mode, also lists dogs and brahmins among playable races.