Ok, they are actually as simple if not more simple than a manual transmission. Their advantages are that they can keep the engine in it's powerband the whole time. For instance, if your car makes max power at 3500 RPM, the tranny will keep the engine at the RPM the whole time your accelerating, creating drastic acceleration increases because your car now has a "flat" power curve. At max power all the time when WOT.
There are all kinds of CVTs, some are hydrostatic (pretty complexed) others are a "belt" or "metal-belt" between two pulleys. One of the pulleys has an adjustable width because it has a spring on one side that can be pushed out. As the speed of the vehicle increases, the "belt" slides down in between the pulley even further as the pulley pushes out further, increasing the gear ratio. Understand?
They can give you better fuel mileage and all around better performance. They are what cars will have in the future for the best performance.
Only bad thing is, the constant "WHAAAAAAAAA...." of the engine won't change (no shifts at all).
But I have ridden a pretty power twin cylinder ATV with a CVT on it and it pulled very hard the whole time on the throttle because it was constantly in it's powerband, all the time, no doubt at all. No more "I wish I had a gear between this and this". It's a continuously variable amount of "gear" ratios.
This page has a pretty good explination of how the CVTs work.
http://www.edmunds.com/ownership/techcenter/articles/45104/article.html