All of these words are used to describe the foot of different animals. The foot is what the animal stands upon. Hoof - foot of the cow, deer, elk, sheep, etc. paw - foot of the cat, bear, dog, etc. claw - claws are sharp and grow on the paws of cats, dogs and bears, etc. The claws on the foot of a bird are called "talons".

Thanks Allen, you're not just talking about language but also talking about the meaning behind the words.

To add to Liberty's answer, as humans, we start out assuming all land creatures have feet that consist of five distinct toes with nails. So, all land animals have feet. However, certain land animals do not have five distinct external toes as some have been fused together or covered by a single hard surface. When this happens, the animal has hooves (the plural of hoof) which is what the hard surface(s) is called. Certain animals with distinct toes have very long, sharp and, usually, curved nails. These are called paws and the entire foot of any animal with paws is called a claw.