Ye Noble Science of Defence
Well - that's what it used to be called. It's been a Science, an Art, a Sport - all sorts of things. Nowadays, the trendies are even calling it the "European Martial Art". It goes from playing with wooden or rubber swords right up to the intricate foot and bladework of the olympics, taking in Basil Rathbone, Errol Flynn, Scaramouche, Cyrano, and loads of other chandelier-swinging candle-chopping loonies on the way.
But that's only my version. Here's the semi-official FAQ.
The Modern Sport
The first and probably most common is the foil. This started out as a practice weapon, to teach the defensive part of sword-fighting. It is the weapon on which most beginners start, because even if they eventually move to another weapon, foil fencing contains many actions which are also relevant to the other weapons.
The epée is a much heavier weapon than the foil. Again, hits are scored only with the point of the weapon, but the whole body is counted as valid target. Unlike foil, there are no rules determining "right of way". 
Sabre is a weapon which evolved from the cavalry sabre, and so has a few differences from the other two. Because of its origins on horseback, anything above the waist is counted as valid target area.