Compound Bow

This type of bow uses a levering system of cables and pulleys to bend the limbs. Eccentric cams reduce the load that the archer holds at full draw which means
they can be held in the aim longer allowing the archer to better compose the shot. Compound bows are very efficient at storing and releasing energy which results in very high arrow speed and a flat trajectory. 

Longbow

The bow of legend, traditionally made from a single stave of yew is now more often made using laminated strips of exotic timber but the archer still uses the back of the hand to rest the arrow as in times past.  Still very popular across all archery disciplines Longbows require much practice , but are very rewarding.

Recurve/ take down bow

Usually refers to a bow with two separate limbs that attach to a handle (called a riser)
used across all archery disciplines there are many variations.