'Purr' is the soft sound (like a car engine running quietly) that cats make when they're contented. You can feel the vibrations when they make the sound. We mainly use it as a verb (to purr). 'meow' or 'miaow' is the noise that comes from a cat's mouth, and the spelling is trying to copy the sound. There are more than two spellings, but these are the most common ones. We use it as a noun (the cat's miaow sounded strange) or as a verb (the cat miaowed loudly). 'mew' is a very similar verb to 'miaow', but it is generally a soft, high crying sound. Kittens often mew rather than miaow. (Just to confuse things even more, we sometimes say 'mewl' ).