Animal Rights, a term introduced by Tom Regan, is the idea that we have some sorts of moral obligations towards animals. So for example, pigs should not be kept in cages to satisfy our taste buds. One very good introduction to A.R. is the Animal Rights FAQ, to be found in the Frequently Asked Questions
One doesn't, of course, have to agree with theories of natural rights to support the basic idea of animal rights on a practical level, just as utilitarians can support various human rights like the right to freedom from torture - although "animal liberation" is a preferred term for some utilitarians e.g. Peter Singer.