The three categories of wheelies:
1) Unweighting or lifting the front wheel. It's quick, not very aggressive, and very useful. The front wheel barely comes off the ground, but it's useful for getting over logs and the like. It's not particularly fun, and can be done with a quick weight shift and a little throttle on almost any bike.
2) Sustained power wheelies. You've got the front wheel decidedly off the ground, but still a ways from the balance point. You're holding it up by being at full throttle, or close to it. This is very useful for getting through water without getting soaked or endo-ing. It's also really fun on a bike with the balls to hold one for a while.
3) Balance wheelies. These are harder, and completely useless. They are stunting. They are riskier to do than a power wheelie, but can be done at lower speeds, since you have to accelerate hard to hold a power wheelie. I can't really do these very well yet, but I keep messing around because it's a fun thing to try and get. I only do this on my dirt bike, because I feel like looping it is not unlikely.