Most "variated" transmission mopeds use a Variable-diameter pulley (VDP, or Reeves drive) type Continuously Variable Transmission.
The drive pulley is often referred to as the "Variator". One cheek of this pulley is static, while the other can slide towards or away from it on a bushing. The variable cheek contains a system of weights, which when acted on by centrifugal force will push the pulleys together, pushing the v-belt out toward the edge of the pulley, and effectively decreasing the numerical gearing ratio versus the driven pulley, and putting the bike in a higher "gear".
The driven pulley can be variable, with a spring pushing the cheeks together to keep the belt riding on the outside of the pulley until the force of the weights overcomes the force of the spring. The spring will then collapse, allowing the cheeks to spread and the belt to ride closer to the center, effectively lowering the gearing ratio. This is the case on bikes such as Vespa, Honda, and Derbi.
Another common solution is to use a static driven pulley and allow the engine to move and release or take up the slack in the belt as the drive pulley diameter changes. The engine is sprung, and the weights have to overcome the spring tension. This can system can be found on french bikes such as Motobecane or Peugeot.
The balance of the forces between the weights and the counter-spring is what determines at what RPMs the transmission variates.
The lighter the weights, the higher the RPM that it will take to produce enough centrifugal force to variate (given the same counter-spring). The heavier the weights, the lower the RPMs.
Similarly with the spring; The lighter, or weaker that it is, the lower the RPM/weight it will give at, the heavier, or stronger spring, the higher the forces (RPM x Weight) to overcome it.
Since it is a crucial part of the tune, for the purpose of this article, We will consider the counter-spring a tuneable part of the variator.
The clutch can be designed in to either pulley, with dual variable pulley systems it is usually on the driven end. With single variators the aftermarket designs are more varied. Clutch tuning is is the same as most bikes, make it grab when the power hits.
For some other explanations, including some great diagrams and animations, check out the links below: http://auto.howstuffworks.com/cvt2.htm
Basic Variator Tuning:
Everybody wants the highest top speed with the best acceleration right? Luckily You get a continuous gearing range to play with, get to tweak where it lies with the final gearing (see below), and decide when You want it to start shifting! The idea is to get the lightest weight and spring combo in there You can while still having enough weight to variate all the way at peak power RPM. That will ensure You get in to the power-band earliest, and stay in it longest. Ideally You want to start variating right when You hit peak power, and stay at that RPM as You variate and accelerate all the way to top speed.
When the weights are too light, or the spring too heavy, the transmission will not variate fully in to high 'gear'. If the weights are too heavy, or the spring too weak, it will variate in to high 'gear' too early, robbing you of power during acceleration, and possibly not allowing the engine to reach peak power RPM.
Now You're ready to play with some weights and markers. Mark up the variable pulley cheeks real good. I like to do this while the moped is running, that way it can warm up, and I have the best chance of a sharpie getting launched in to my eye! Now You can tell how far it variates while You're riding without looking down the whole time and running off the road in to a ditch. Go for a ride and check how far up the pulley the belt rode and rubbed off the marker.
If the bike feels ok, You are reaching good power RPMs, and the belt rode all the way up to the top of the pulley (usually the case with stock vario and tuned up engine) You are ok, but could possibly gain some acceleration by using lighter weights or lightening the weights in the variator. You can try lighter ones until the belt no longer reaches as far to the outside of the pulley as it can, then go back to a little heavier, from there You can fine tune for Your powerband. Even if it variates fully but too early, top speed can be lost.
If the belt did not ride all the way up to the top of the pulley, the weights are not heavy enough to variate at the RPMs that You are reaching. One of two things could be happening; if You are hitting target RPMs they are simply too light. If You are getting in to the powerband late, or not at all, then the weights are too heavy and Your bike is not able to pull the effective gearing.
If You have an assortment, it is generally best to start with the lightest weights, or ones that You're pretty sure will be too light, and work your way up to full variation.
When even the lightest weights won't do, a heavier spring can be used. This is usually avoided on dual variable pulley systems by people who want to keep the mass of the rotating assembly on the crank shaft to a minimum. It can be unavoidable, however, if the stock spring is too weak to keep tension on the belt and push it back to the outside of the pulley. This can allow the transmission to stay in a higher gear when starting up a hill or after slowing down, the variator to fluctuate or 'bounce' on acceleration, or even the belt to slip.
Advanced Variator Tuning:
For this You will need a tachometer and knowledge of where Your engine makes the most power. Someone might put more details here, but If You're this far You probably already know what to do. Here's a really great thread that will give you insight into your next steps.
There is usually a final gearing system of some sort employed between the CVT output and the wheel. In the case of Dual-variable pulley systems it is usually a transmission contained within the rear wheel hub. On the french bikes it is a chain and sprocket system. The idea here is much the same as with any other bike, You can gear for acceleration or gear out until the bike can't pull any more for top speed.
Vespa Variator Tuning:
Peugeot Variator Tuning:
Peugeot stock variator tuning - a story in pictures by Graham Motzing
Stock Peugeot Variator Mod: How-To by Jake P