What is the difference from tennis shoes/gym shoes and running shoes?

i have a pair of pumas, but i think they arent meant for running. i think they are meant for playing tennis or something like that. but i have joined track and i want to get running shoes. how can i tell if my pumas are running shoes? because i want to make sure they arent before i buy running shoes. and when i buy running shoes, i want to make sure i am actually buying good running shoes. k thanks (: