An exploit is a kind of bug. Generally, bugs are things players hate and can ruin a game. Exploits, however, are those rare few bugs that actually benefit a player. For example, being able to duplicate money or items, or becoming invincible under certain circumstances. In single player games, using an exploit isn't much different than using cheat codes, however, in multiplayer or MMO games it can be a serious offense. In games that are watched by admins or GMs, being caught exploiting a bug can result in a ban from the game.
The proper response when finding an exploit in a multiplayer game, is to submit a bug report to the game developers and refrain from using it, or telling others how to use it.