A minigame is a (usually short) segment of a video game that uses a different style of gameplay than the rest of the game. For example, an adventure game may have a boxing minigame, or a third person shooter might feature a racing minigame.
The use of minigames varies greatly by game and by genre. In simulation games, minigames are intended to mimic a certain aspect of the simulation (for example, the hunting sequences in The Oregon Trail). Role-playing games often include complex, optional minigames that unlock a hidden feature of gameplay (usually granting the player's character some new ability). Some games, particularly anthology-like puzzle like Wario Ware, are made up of minigames so short they can only be called Microgames, and games like Mario Party are quite specficially a bunch of minigames strung together.
Another example of a minigame is Arcomage, a game from Might and Magic VII. It involves building up your tower whilst trying to destroy the computer opponent's tower. It's a simple but very addictive game that could be played at the inns around the Might and Magic VII game-world.
In the latter case, they are often called "secret games." In the former case, the successful completion of such minigames may or may not be required in order to finish the encompassing game.
Similar games such as the Mario Party series are considered party games, games specifically developed for multiplayer amongst players.
These games must be completed in order to yield the Nintendo and Rareware "coins", tokens which unlock the final area of the game.