CVT transmissions can work a number of different ways but the most basic method is having a set of pulleys and belts that shift up and down on a shaft to alter the angle at which the belt is sitting on the pulleys. This changes the ratio from input to output based on how the pulleys move on the shaft, it is important to note that this doesn't affect the distance or length of the belts in anyway. The big difference between the two is CVT transmissions allow constantly variable ratios meaning they have no fixed gears like a traditional transmission. This allows them to increase fuel efficiency by keeping the engine at it's sweet spot all the time. This also means in theory the car would also accelerate fast due to the engine remaining in peak power all the time. A traditional automatic works based on fixed ratios and has a set number (5 speed for example) this means when the transmission changes gears there is a drop (or increase) in engine revolutions which could take it out of peak power or put it in to peak power.
In actual practice the CVT isn't as great as I had thought, the fuel efficiency would be nice but getting it isn't worth the sacrifice in my mind. I think I now know why luxury manufacturer's like Nissan's Infiniti use 7 speed autos instead of the CVT, it's very loud and seems very unrefined as a result. I wonder if soundproofing could solve this issue and a large displacement engine would allow the car to rev lower during acceleration and thus create less noise. I'm guessing it would help to a certain degree but not enough otherwise we would see Infiniti's with CVTs.