Rims are the metal support for the rubber of the tire, and means of conntection to the axel. Instead of being a weighty piece of metal, the rim is composed of spokes which connect the outside portion to the mounting area in the middle. The number and thickness of the spokes vary by design, and are usually designed to be visually appealing as well.
Performance oriented rims are almost always designed to be lighter than standard ones, since they are part of the unsprung weight. Some of them are also designed to be aerodynamic, or to draw in or blow out air in some special racing vehicles. Performance rims often cost more since they have to made out of strong, lightweight materials, and have strong archetexture to allow as little material as possible to further reduce the weight.
The bead of the rim is where the edge of the tire sits on the rim. Some rims have a flat surface at this point for weight attachments to be added for balancing.