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A Skybox is, in a nutshell, a graphical representation of the sky that 'wraps around' a computer game-world, and is used to simulate the 'sky' of a game-world. In most cases, a skybox will contain the typical aspects of what constitutes a 'sky' — clouds, the sun, and (occasionally) birds and other avian life. Such life is usually non-interactive, and appears either as frames of static animation or as randomly generated graphical elements.

The use of a Skybox should be apparent: It allows the programmers to use relatively little memory to simulate a functioning day/night cycle and atmospheric system, even within the incredibly confined and compressed geography of a computer game (where the actual, physical distance between ground and sky may equal as little as a mile).

Skyboxes are independent graphical elements. Usually, a given skybox can be 'swapped out' for one that suits a particular setting. Some computer games come with several Skyboxes, showing different types of meteorological phenomena (such as aurora borealis or other things that can't physically be rendered) or a even an entirely different setting (apocalyptic vs. modern, for example).

Note that Skyboxes are specific to computer games; animated cartoons don't use Skyboxes, for the simple reason that there's no need to use them - the sky will naturally be drawn as part of the cartoon itself.

The Skybox may also contain 3D models that will be projected larger than life, such as distant land features or buildings— in fact it usually does, in newer games, since it's a great way to make the world appear richer. It may even contain objects that move and change, such as the Citadel in Half-Life 2, or even animated characters (who will appear to be distant Godzillian monstrosities, something occasionally used to good effect in Machinima.)

Examples

Used in Sonic Adventure 2. The levels look like they occur in the real world, but there are glitches that allow the user to jump out of a level. Doing so reveals that the levels are hallways built in a large skybox.
In Spyro the Dragon, flying outside the bounds of the flight levels reveal that you are inside a skybox.
One of the maps in Team Fortress 2, Well, uses the skybox to conceal the interiors of the teams' bases, which are Bigger on the Inside. The top half of the buildings' fa�ades are 3D miniatures projected onto the skybox. Generally this is not feasible in the game's engine, since the technique requires that there not be a direct line of sight to any part of the area being hidden.
In Mabinogi: Fantasy Life, the skybox moves as time flows; since time is sped up in the game compared to real life, you can actually watch the clouds/moon/sun move across the sky.
A toggelable option in Runescape. Used most notably in the Clan Citadels.
Roblox has tons and tons.
Modern Game Engine come with built-in skyboxes, an example is Unreal Engine from Epic Games. It's most likely the first thing you see when you load up the development environment.