A triathlon is actually three events formed into one race, swimming, cycling and running.
The basic premise of a triathlon is to swim for a while, cycle a bit, then run for the finish line, doing all as fast as possible.
Triathlons originally started in America as a form of cross training to keep a group of runners from getting bored when they weren't racing. It slowly spread from there and is, for the first time, featuring in the Olympics at Sydney 2000. Of course, whenever you mention triathlon to someone, they may often say "oh, that thing in Hawaii?" Well, yes, but that is the Ironman, which is very different and far further. This was started as a bar bet between a group of men, one of whom was John Collins, the founder of the Ironman race. There were, at the time, a sea swim of 2.4 miles, a long bike ride of 112 miles and finally the marathon of 26.2 miles. John Collins said that, to work out who was the fittest, they could do all three. One after the other.
There are various distances, ranging from short distances for novice races, often aimed at beginners, to the standard races of sprint distance and olympic distance, through every distance right up to the ultimate, the Ironman. Depending on the location and length, the swim section is often held in open water, a lake or even the sea. Of course, in the cooler areas of the world, wetsuits are pretty important for this stage.
A sprint distance race is usually 750m swim, 20km bike ride and 5km run. The Olympic, or international distance race is 1500m swim, 40km bike and 10km run with the top athletes finishing in approximatley two hours.
Often considered the ultimate trialthlons are the Ironman events (history above). The record for this is about 8 hours.
However, as long as you can float, ride a bike and jog, then triathlons are one of the best ways to get fit, meet people and have a good time. You never know, you might be the next olympic champion.
In the UK, the main organisation is the http://www.britishtriathlon.org British Triathlon Association, who sanction races, set the rules and generally organise things.
For UK web heads, a good starting point is the http://www.triathletes-uk.org triathletes homepage who have links to almost everything you could want.
Internationally, http://www.triathlon.worldsport.com Worldsport's Traithlon page is a good start. Any decent search engine will have hundreds of links.
Get out there, give it a go and have a good time.