Prong, channel and bezel

Prong setting
Metal shaped into claws grips the diamond, showing its brilliance from every angle. Classic styles are named by the number of prongs: three, four or six.
Channel setting
Small diamonds sit between two parallel strips of metal. It is mostly used for pieces whose main stones are small, or that have no center stone.
Bezel setting
A rim of precious metal encloses the diamond below the girdle in a metal mount. The base can be closed with metal or left fully open, and a bezel-set diamond is very secure.

Tension, bar and pavé

Tension setting
Also called a clamp or pressure setting. The metal presses inward on the diamond's girdle and holds it by tension, so the stone seems to float in the metal.
Bar setting
Also called a shared-prong setting. Fine metal bars hold each diamond separately, or two neighboring diamonds share a pair of bars, for a look that is rich yet delicate.
Pavé setting
Also called a bead setting. Small beads are raised with a tool from the metal at the edge of each seat to hold the diamonds, and the setting also adds sparkle to the center stone.