From the Greek words auto- (meaning 'self') and tomos (meaning 'cut'), this refers to the ability of some animals to 'break' without causing serious harm to themselves. The most widely recognized examples are lizards which can loose a tail when attacked.
Many animals have the ability to break themselves; this is most often done in self-defense or as a means of hibernation and reproduction. I will list some examples below. In all the cases given below, autotomy is accompanied by regeneration of the lost body parts.
Vertebrates: Reptiles
The best known type of autotomy is probably the caudal autotomy of many lizards. Some types of lizard can 'drop' their tail when they are threatened with attack, leaving it as a twisting, thrashing distraction while they either escape or freeze in an attempt to escape notice. Salamanders can also autotomize their digits and limbs (if your lizard is loosing digits, however, you need to get him to a vet).
Some lizards can only loose a tail after a predator has grabbed it; this is also usually referred to as autotomy, even though it isn't truely 'auto-'. Some skinks and geckos can 'drop' patches of skin in this way.
Arthropods: Crabs
Some species of crabs can drop claws and arms in much the same way as lizards can drop their tails. This can be done to help them escape, to reduce bleeding in an injured extremity (loosing an claw or arm causes a large blood clot to form almost immediately at the break point), or to rid the animal of a diseased claw.
Annelida: Worms (blackworms, polychaete worms, etc.)
Some types of worms will autotomize their bodies; each piece will develop in a whole worm. (This is a form of asexual reproduction). This is more extreme than either of the previously given cases, as any body part, including the head, may be regenerated. A worm may autotomize spontaneously; it may also do so in response to stress, such as a predator attack. In the laboratory, compression of the body is commonly used to stimulate autotomy.
Echinoderms are also known for their ability to autotomize.