A minigame can be described as a videogame within a videogame. These minigames often have a more simplistic nature then the game it comes with and is meant to serve as a level filler, something to pass the time while a portion of the game is loaded, as a reward for finding easter eggs, but are mainly meant as a fun distraction from the gameplay offered by the main game.
One of Final Fantasy VII's many mini games (more then 30) allows you to enter a chocobo race.
Minigames come in a large variety of forms and lengths, ranging from puzzles to slot machines to shooters. They are also presented in just as many ways within a game, such as an in game console or your games organizer such as PDA's.
Who hasn't played Doom 3's Super Turbo Turkey Puncher 3? Made by Nabcon!
Besides acting as a distraction from the main game's gameplay, minigames can also serve as a means to progress the main game itself, such as picking locks to a chest or door, hacking a computer to access it's data, cooking up a potion to restore your health or creating that fie piece or armor or weapon to help increase your chances of winning. The successful ending of these minigames often depend on the character's skill in that area and if completed successfully often lead to hidden treasure and a shortcut to advancement.
In Elder Scrolls IV: Oblivion, picking a lock can lead you to some interesting loot... and sometimes into some nasty fights with the city guards...
Minigames have been around almost as long as videogames itself and have become increasingly popular over the years. Some minigames have even become so popular that they have become standalone titles, such as Geometry Wars, which was originally a minigame in Gotham City Racing 2. Some games, such as Wario Ware or Mario Party, are actually a compilation of several minigames.
Mario Party has always been full of memorable fun minigames.
Nowadays minigames are a booming, million dollar business, no matter which platform you own. Online platforms, such as PSN and Xbox Live, are literally flooded with purchaseable minigames and games themselves are sometimes teeming with them. Any way you look at it minigames have become an integral part of games and the gaming industry and will (hopefully) continue to do so for a long time to come.