They are similar, in the same way that friend and buddy are in english. Ami is a little more generic, and copain a little more familiar or affective. But really they are equivalent words. Feminine : amie and copine. Ethymologically ami is someone you like and copain someone with whom you share bread. So generally copain is a better/closer friend than a standard friend, while ami is generic and can mean both. As stated by Jojo and Anatheme, petit-ami and petit-copain are translations for boyfriend, but note that the sole word copain depending on context can mean that too, so be aware there is potential misunderstanding.

pour moi un amis i's a friends ( best-friends) et un copain is just a friend par contre petit-copain ou mon ami is a boyfriend

Ami is friend. Copain can mean either friend or boyfriend: usually mon copain = my boyfriend; un copain = a friend.