A trackball is a computer pointing device that is a lot like a mouse turned upside down, albeit with a much larger ball. Normal mice are rolled around on the desktop, while a trackball remains stationary, and instead you roll the ball around.
Trackballs versus mice is usually a personal thing. Some people say that trackballs allow for greater control, while other people say that they can't use the damn things at all. I myself can use them interchangeably with mice, and think that they are much better for gaming then mice ever were.
One major advantage of the trackball is that you don't need a large flat space to use it on, you can put it just about anywhere. Many people with certain disabilities find a nice heavy trackball much easier to manipulate than a mouse.
The trackball actually saw its first widespread usage in the arcades, and not on the desktop. Atari first used the trackball as a control mechanism in their Atari Football title. Trackball based games quickly became popular, and have remained so ever since. Titles such as Missile Command, Centipede, Crystal Castles, and even the latest Golden Tee Golf all use trackballs as a primary controller.
Both arcade and computer trackballs function in pretty much the same way. The ball turns a pair of optical encoder wheels, which is then translated into movement by the computer. That is how most trackballs work, some of them use a laser instead, like the Intellimouse does. Arcade trackballs are almost universally compatible with each other, and can be used on a PC if you do some custom soldering, or order an interface board. While computer trackballs are widely available in PS/2, serial, and USB versions. Some ancient ones even shipped with their own proprietary interface card, but you can't buy those anymore.