Possibly by 1500 BC people in Mesopotamia used rope pulleys for hoisting water. Archimedes of Syracuse invented the first compound pulleys 287 BC - 212 BC. Plutarch reported that Archimedes moved an entire warship, laden with men, using compound pulleys and his own strength.
There are different types of pulley systems:
Fixed or class 1 pulley has a fixed axle. That is, the axle is "fixed" or anchored in place. A fixed pulley is used to change the direction of the force on a rope (called a belt). A fixed pulley has a mechanical advantage of 1. A mechanical advantage of one means that the force is equal on both sides of the pulley and there is no multiplication of force.
Movable or class 2 pulley has a free axle. That is, the axle is "free" to move in space. A movable pulley is used to multiply forces. A movable pulley has a mechanical advantage of 2. That is, if one end of the rope is anchored, pulling on the other end of the rope will apply a doubled force to the object attached to the pulley.
Compound pulley is a combination fixed and movable pulley system.
Block and tackle - A block and tackle is a compound pulley where several pulleys are mounted on each axle, further increasing the mechanical advantage.
Sources: Wikipedia, Mechanics By Henry Kater