For nearly all applications, motors spin faster than is optimal. Although speed can be important, in most cases the added torque gained via speed reduction is needed to get the performance you are looking for.
Optimizing Drive Speed
How fast should you be? Speed vs. Pushing Learning to drive Robotic Legend has created a nice tutorial on how to calculate speed.
Optimizing Weapon Speed
Spin up time is as important, if not more important than how fast a spinning weapon spins at top speed. If your weapon takes too long to spin up, an aggressive opponent will make contact with your slow moving weapon without taking damage. Typically you want to be able to reach 90% of your top speed in 3 seconds or less. Quick spin up times can be even more important in the small weight classes where small arenas mean you are never far from your opponent.
In some smaller robot classes that can get away with direct drives for their weapons but typically the motor shaft won't be robust enough to survive weapon impacts. In most larger robots and many small ones, direct drive weapons are too fragile and you will want to transfer the motor speed to a weapon shaft via gears, belts, or chains. This transfer of power from the motor to the weapon-shaft is an opportunity to adjust the weapon speed to find the right balance between spin-up time and top speed.
Faster is always better isn't always true. Although high speed is needed to make the robots weapon destructive, spinning the weapon too fast has some downsides. Gyroscopic forces can make turning the robot difficult, and some weapon designs like drumbs or disks may have difficulty 'biting' into an opponent. Also, some events or robot-classes such as sportsman robots may have rule limitations on weapon top-speed.
Gears are wheels with teeth that slot together. When one gear is turned the other one turns as well. If the gears are of different sizes, they can be used to increase the power of a turning force. The smaller wheel turns more quickly but with less force, while the bigger one turns more slowly with more force. Although rarely used in weapons, gears are a popular choice with drive systems as they come in a variety of sizes and configurations.
Pros: Gears transfer power efficiently.
Pros: Gears can be arranged to take up very little space inside a robot.
Cons: The build quality of inexpensive gearboxes rarely hold up in combat robots.
Cons: Gears are heavy.
A Roller Chain, known as the drive chain or transmission chain, passing over a sprocket gear, with the teeth of the gear meshing with the holes in the links of the chain.
Pro: Chains are a highly efficient way to transfer power.
Pro: Physically strong.
Cons: Often requir exact spacing or tensioners to keep them on their sprockets.
Cons: Chains are relatively heavy.
Drive belts use friction to operate. Tension is applied to the belt which causes friction between the belt and the pulleys that it's installed around. Some belts and pulleys have 'teeth' or ridges to increase friction and keep their motion in synch.
Pros: Belts are a light-weight solution for transfering power.
Pros: Belts come in a wide variety of sizes and styles making them suitible for many designs.
Cons: Belts can break in high energy applications.
Cons: Belts can loosen over time which reduces their effectiveness.
Friction drive is a system that depends upon friction between moving parts in contact to transmit motion. Often a wheel is directly attached to a drive motor which is then pressed against another rotatiting part to impart motion.
Pros: Friction can be simple to impliment
Pros: Friction drive can be light weight since it doesn't require heavy parts like gears or sprockets or chains.
Cons: Maintaining the right level of friction can b challenging.
Cons: Friction drive wheels can wear out quickly.