Angora seems synonymous with soft and silky; it's a type of long-haired cat, goat, and rabbit, as well as a wool made from the rabbit fur. That wool makes a lovely soft sweater, much prized by bad movie director Ed Wood and inspiration for the clever name for a movie about him, Look Back in Angora.
The cats are an ancient breed of medium long-haired cat which, unlike Persians or Maine Coons, have a single coat with no undercoat, making them easy to groom. They come in many colours and are known to be intelligent, even manipulative. In the way of such things, there's an exhaustive list of breed qualities which a "show cat" must attain; see the list at www.palantir.co.uk/ang2.html.
The goats originated in Asia and were bred for their production of mohair, a smooth thin fiber that is similar to wool, but more elastic and without wool's felting properties. Mohair dyes well and is mostly used for making plush upholstering material which is strong and durable. The goats are raised around the world and sheared once a year. For more info on these guys, see www.ansi.okstate.edu/breeds/goats/angora/.
The rabbits are white with floppy ears and fluffy bodies, and their coats are sheared throughout the year to get the wool to make a sweater. (yclept says that English angora and French angora rabbits are also "plucked," where the fur is pulled gently, and the resulting fleece is placed between sheets of tissue. This receives a higher price than sheared fleece, as it's only the longest hairs.) It can take one rabbit a long time to produce enough wool for a sweater. Some people raise these critters for wool, while some have them as pets. There's a long FAQ on angora rabbits and how to breed them at www.angorarabbit.com/angora-rabbit-manual/.