Recurve is the most commonly seen bow style as it is the only style that currently competes in the Olympics. It consists of a riser handle), limbs, string, a sight to aim with and stabilizers to help keep the bow steady.
Barebow is a stripped-down form of Recurve that features no sights, no stabilizer, and no markings on the bow to help you aim.
Compound requires a very different setup to barbow and recurve. One of the main advantages to compound is the apparent reduced bow weight. The string on a compound bow runs over two cams which give the bow a "valley". This means that once the bow is drawn back to a cretain length, the draw weight of the bow "drops off". Another aspect of compound is that the sights that are used in compound can be magnified. For a lot of these reasons, compound is considered the most accurate of the three bow styles.