First muscle fibres. The best way to think about the differance between fast and slow twitch is to think about the differance between a 100m sprinter and a matathon runner. The sprinter has large, powerful legs, capable of launching him/her with incredible power and speed, but wears out quickly. The distance runner has slender, less muscular legs, but is capable of going on and on all day without growing tired. Another way to think of the two is the differance between a rocket and a jet. Both consume fuel, in this case oxygen and sugars. While a rocket can produce amazing power, it only operates off a set amount of fuel, and is thus limited to its performance over time. Once the rocket runs out of fuel, it needs to be recollected and refilled, which takes time (longer recovery) A jet on the other hand, consumes a refillable fuel, does not burn itself out as quickly, and can be refuled much faster, or if operating at a low enough capacity, wont require a stop at all.
Both muscle fibres have thier place, and you need to think about what your needs are in life when training for them. If you need to have great strength, but no need for continuous output, train for fast twitch. if you need less total power, but want to go all day, train for slow.
The other peice to the puzzle are the neuro-motor controls to your muscles. These tell the fibres when to fire, and how hard to do so. By maximizing your motor control, you maximize your output. think bruce lee. Small guy, but incredible power. How? By mastering his motor controls he was able to make as many fibres fire up when he needed them as possible.
As a general rule of thumb, if you train in the 1-5 rep range, failing on the last rep, you are training the motors. train in the 4-12 range and you are hitting mostly the fast twitch. train in the 10-unlimited range, and your hitting first your fast twitch, then as they run out of fuel you rely on your slow twitch.
Its important to note that you cant effectivly train just your slow twitch fibres. during the first 4-12 reps, both fast and slow twitch are doing work. thus, when building muscle you will typicaly build more fast than slow twitch.
My own training system, when i'm being good and practicing what i preach, is to spend time split between trying to train my slow twitch fibre and motor controls. My theory is, by mastering the motor controls and strengthening fibres that dont tire, i get the best of both worlds. incredible power, and endurance at the same time.