Its a lot more complicated than what I'm about to post - but I'll try to keep it short.
Fast twitch muscle fibres use an anaerobic method of producing energy to contract the muscle - the system for doing this is quite quick and hence the muscle can be contracted quickly - however the process also produces a lot of byproducts that block the receptors that cause contraction - thus the muscle fibre fatigues quickly.
Slow twitch muscle fibres use an aerobic method (ATP transfer) to generate the energy for contraction and produces less by products than the anaerobic method. This means the muscle can contract over a longer period without fatigue however the ATP cycle is quite long compared to the various anaerobic processes and hence the muscle cannot fire quickly.
I've not read any of the latest research but about 2 years ago when I was reading through stuff about fast/slow twitch fibres there was a raging debate about whether the fibre percentage was purely genetic or could be altered with training. Considering the way the body works, I'd say training would affect the percentage but genetics also plays a role.
Basically if you seem to be good at endurance sports then you probably have more slow twitch, if you are good at explosive force stuff then probably have more fast twitch.