In the purest sense, a supercharger is any device that provides the engine
with a greater charge of air
(not fuel) than the cylinder
can could intake through the down motion of the piston. It is a pump
that specializes in pushing air into the intake, while the engine itself is not so specialized.
In practice there is a huge division: turbochargers are superchargers that use energy from the exhaust gases to drive a turbine which drives another turbine that pressurizes the intake. Basically every other device that does the above but isn't a turbo is called a supercharger.
Even more generally, when people say "supercharger" they are generally referring to a particular type of supercharger, called a roots-type supercharger. This type of supercharger uses two rotors with lobes that spin together pushing air into the intake. It is driven via a belt, and its spin is proportional to the speed of the engine's rpms, unlike a turbo which is more proportional exponent of the engine's speed (well, not really, but for our purposes, pretend...). This leads to a difference in the response of the roots-type supercharger and a turbo. A roots-type supercharger gives the engine a boost in the lower range, and raises peak torque (which is often used to express the low rpm power of the engine) but near the top end, the supercharger is inefficent. A turbo doesn't really give a large boost in the low range, because the turbo must spool, leading to the infamous turbo lag. However, once the turbo is spooled up, it is a much more efficent design, and raises the top end and horsepower signifcantly more than a supercharger.
A supercharger cannot reach the lofty heights of a turbo because when spun at high speed, the supercharger causes a huge amount of heat, a result of concussions as tiny packets of 1 bar (atmospheric pressure) air smack into a 1.75 bar mass of pressurized air. A supercharger doesn't have "internal compression," that is, it doesn't compress the air as it is running through the device. Hence, it is also called a blower. The high temperature of the charge resulting from a supercharger at high rpms is very very bad. It is the destroyer of engines, the ruiner of cars, and the sacker of races. Hot air increases your chances of detonation, or preignition.
Most people say that a supercharger will not go over 15psi (or approx 1 bar), this is probably true, but even at one bar, the supercharger is generating so much heat, it's probably doing more harm than good. It would be a good idea to use an intercooler or some other charge cooling device (or a combination) if you really want to run a supercharger at high boost.
One of the biggest manufacturers of roots-type superchargers is Eaton.
All the the previous, of course, only applies to roots-type superchargers. There are several other types (besides turbos, which are generally segregated): centrifugal, lysholm, and roots are probably the main types used today.
A centrifugal, also known as a Vortech (after a company who makes them) is about as different from a roots as it could get, and despite that people often call it a supercharger, suggesting that it is a roots type. A centrifugal uses a turbine mated to a belt drive or a electric motor drive. This gives you the downsides of a turbo and and the downsides of a roots all in one. First of all, it is driven proportionally to the engine's speed, so you will never get the exponential speed curve that an exhaust driven turbine affords you. Second, you are using a turbine (at either too high or too low of a speed) so all your gains are going to be biased to the top end (but you are limited because of the linear speed curve). Bosch suggests a continuously variable transmission to allow the turbine to spin at the appropriate speed. Why not just get a turbo? The best success I have seen with these is on very torquey truck engines, as they will give a little boost up top.
A lysholm, aka whipple or whipplecharger (again, after a company that makes them) is sort of a modified roots. It uses a set of screws that compress the air as the travel up the screws to make up for some of the inefficencies of roots-type superchargers. They are also easier to spin faster. They are very common in Australia and Europe, but rarely seen in America and Japan.