A climactic confrontation with a unique, stronger than average, and often monstrous enemy creature known colloquially as a "boss". It has several important features, depending on the genre of game in which it is found.
In old-school Shoot 'Em Ups and 2D Platformers, the boss comes at the end of a level or series of levels (commonly called a "world"). He will often require a special trick to defeat, or have a targetable Weak Spot.
In FPS, 3PS's, and 3D platformers, they divide up the game. They might be common or uncommon, gigantic or small, complex multi-stage affairs or simple exercises in circle strafing while shooting.
In RPGs, they wait at the end of dungeons (though the larger dungeons may have multiple boss fights). They have a mountain of Hit Points. If they're the last one, you'll most likely fight them more than once. Most have a specific weakness.
In a Fighting Game, the boss is simply the last character you fight, in that even when the rest of the battles have you going up against a randomly-selected opponent, the boss is always the same. The boss is typically not a playable character (though circumstances may allow him to be unlocked later), and tends to be the most difficult opponent. Occasionally, the penultimate battle will also be against a "boss". (The penultimate opponent is often dubbed a "Mini-Boss".)
Note that in some games, the boss isn't always at the end of any given level or dungeon.
Tropes:
Advancing Boss of Doom A boss that keeps chasing you and you have to continually flee from.
Go for the Eye If the boss has a giant eyeball, nine times out of ten that's its weak point.
In Case of Boss Fight, Break Glass If the boss has a glass cockpit, frame concealing energy core or generally any other glass part on the boss's body, nine times out of ten that's its weak point too.
Cores and Turrets Boss A completely mechanical boss that mainly consists of turrets (which shoot at you) and cores (which you shoot at).
Cowardly Boss A boss that the party fights, that then randomly runs after taking a certain amount of damage, forcing you to have to hunt that boss down and fight them again.
Crosshair Aware The player will be fighting a boss, and a target sight will be shown tracking the player.
Damage Sponge Boss The most basic boss type: Strong attacks, lots of Hit Points; the only strategy is to hit it until it dies.
Duel Boss A one-on-one boss fight, in a game that primarily involves team battles.
Degraded Boss Once you beat the boss, it comes back as a normal enemy later.
Disc One Final Boss Yeah, that guy who blew up your home? You beat him and he was someone's minion.
The Dragon The villain that's second in power after the Big Bad, and is often the second-to-last boss.
Early Bird Boss A boss that appears early on and is difficult primarily due to your characters' lack of stats and abilities that they will acquire later.
Stationary Boss A fight where the boss remains in one place.
Pivotal Boss The boss is too big to move, so he turns to face you.
Tactical Suicide Boss The boss is impervious to the player's regular attacks, and would be impossible to defeat if one of its attacks didn't expose its particular Achilles Heel in the process.
Boss Arena Idiocy The boss's weakness is something unique to the boss's own arena.
Teleport Spam A boss that teleports all over the place.
Tennis Boss You have to reflect his attacks back onto him to defeat him.
That One Boss An extremely difficult boss that feels hopeless to defeat.
