A router essentially determines which way is the shortest or fastest in a network, and routes packets accordingly. It works at layer 3 of the OSI model, moving packets from one port to another based on L3 addresses - ie. IP addresses, IPX addresses, etc.
A switch connects one point to another in a network temporarily by turning it on and off as necessary. It works at layer 2, with some intelligence (there are also some layer 3 switches, that essentially have routing capabilities).
A bridge connects one point to another in a network. It works at layer 1 and 2 of the OSI model. It only connects two segments of the network.
A bridge and a switch are very similar. In practice, a switch can be looked at as a multiport bridge - both have the same basic functionality. They move frames between ports based on MAC addresses. A lot of functionality can be added to switches, but the base purpose is the same. "Layer 3" switches can be looked at as a switch with a router inside.
For more information on differences, see the OSI model layers here:
OSI model
