Is it illegal to catch a snake and keep it?

It depends entirely on where you live and what species of snake it is. I would not recommend catching a wild animal to keep as a pet. They don't usually adapt well to captivity and it can be tricky to get unhappy snakes to eat. Many of the more common snake species (as well as the more exotic ones) are bred in captivity to be kept as pets. I would buy from a reputable breeder or a pet store that you can be sure is not dealing in wild-caught snakes.