An enemy is any sort of computer-controlled NPC or entity that tries to harm you. They are sometimes called monsters or bad guys. In many games, the common enemies are almost as recognizable and iconic of the series they appear in as the main characters, such as Goombas from the Mario series or Metroids from the Metroid series.
While there is some criticism towards the use of bosses, almost everyone likes enemies, because they're easy to kill, and killing things is fun. And because enemies often drop power-ups, beating them up can be something to look forward to.
Enemies are basically a tool used by developers to keep the game going and pace out levels between boss fights.
As games advance, enemies are becoming more and more involved in the gameplay. For example, in The Legend of Zelda: The Wind Waker, Link could steal weapons from enemies, and lure them away with bait. In Eternal Darkness: Sanity's Requiem, enemies could be 'hijacked' with magic and used to complete physical puzzles that would be otherwise fatal to the player. In Metal Gear Solid and some similar stealth-type games, the focus is not on attacking enemies head-on, but rather sneaking around them (or eliminating them in a covert manner).