A rival is someone who you in competition with and who is about the same standard as you. It can be used, for example, where there are two or more businesses in the same market- Coke and Pepsi are rivals.
An opponent is someone you are directly competing against; in a sports match your opponent is the other player/team. It can also be used for fights.
An enemy is someone you dislike and who often tries to beat you or to prevent you from making progress.