What do they do and if I put one on a quad, will it hurt the quad in the long run?

 

A supercharger is a pump. Nothing more, nothing less. In the engines case, it pumps more air into the engine. The trick to more HP is more fuel and air through an engine in a given amount of time. A non-supercharged engine (NA) can only draw so much air in on each intake stroke. With a supercharger, you can increase that amount to 50 100 200 300+% more air. When this happens, the engine will think it is bigger than it really is.
For example: If I run around 14PSI boost on my supercharged 350CID engine, it now thinks it's a 700CID NA big block engine.

Supercharging an engine adds more parts and these parts can fail. That being said, if a supercharged engine is done correctly they should last a very long time.
There is a member on this form that has a supercharged 700KFX. I have read it is super fast.

 

That's a pretty good explanation.

A supercharger is an engine driven pump that pressurizes the intake charge. The usual configuration is a vein type pump driven by a belt. Pressure in the intake system increases density and flow into the engine, and getting more fuel/air into the engine produces more power.

By comparison a turbo charger uses an impeller to pressurize the intake system, and it is driven by a turbine wheel spun by the exhaust.

Either system will add additional strain to the rest of the engine (as will anything that increases engine output), so it would be important to limit the boost and power increase to what the engine can stand.