The S/C started out on the 1906 Chadwick car and have been used ever since in vehicles

The most common type used commercially and aftermarket is the Roots type
Eaton is the largest manufacturer and uses an M disignation M45/62/90/112 with each number representing the cubic inch of the S/C housing displacement. Basicly it has dual rotors with 3 vanes that forces and compresses the air into the intake manafold or intercooler then into the intake runners to each cylinder. Ford and Mercedes is probably the 2 biggest users of the Eaton Roots type.

Next is the twin screw Kenny Belle and Whipple are probably the most knowed for this type where the roots has rotors spinning and catching the air the twin screw has 2 screws that intermesh and compresses the air and forces it into the intake or intercooler. It also is more efficent at higher RPM's over the Roots as the Roots after reaching a certain speed only blows hot air due to the air being compressed. Whipple and KB uses a Liter disignation on they'er S/C's like a Whipple 2.3 is 2.3L of displacement

Then Centerifugal supercharger is the next, Vortech/Novi/and Procharger are probably the most known in this catagory. It's operation is basicly a Turbo on a belt. the advantage is that there is no turbo lag. And since they'er more compact over the other designs can be used in a tight space like under a hood of a Corvette. It's operation is it takes in air and compresses it in the turbine housing and forces it in the intake tube to the intake manifold. Its only major fault is it doesn't make usable low end torque over the other types so if you use it in a small displacement engine under a 350 in a heavy vehicle 4000+ Lbs it doen't take off real fast over the other types.

Then the oddest one Millercycle engine. Only 1 vehicle ever used this and that was the late model Mazda Millennias. Its a modified 4 stroke engine. During the intake stroke a supercharger (roots type) feeds compressed air into the intercooler then this cooled air is feed directly into the cylinders. The intake valve remains open longer than a normal 4 stroke engine until the cylinder moves 1/5th from top dead center then the valve goes closed and compression is achived. The V6 Miller cycle that the Millennia used was only 2.3L and made 210 HP.