You can tame an animal by having it walk into a trap. This can be easily done by placing a crop next to or on top of a trap.
Only small animals can be caught in small traps, however a normal trap can catch any animal, including large.
If you tame a sheep, you can shear it for its wool with a knife without killing it outright. It grows back.
If you tame a cow, you can milk it with a bucket.
If you tame a dog, you can use it to attack.
If you catch a small animal (pigeon, raven, chicken, jackelope, mouse, ferret), you can perch it on your head.
If you tame a large animal (horse, deer, pig, cow, sheep, goat, dog, cat), you can ride it.