A relay is a switch that's controlled by electricity. It consists of a set of mechanical contacts that come together when the relay's "coil" is energized. The coil creates an elecromagnet that pulls the switch contacts together.
Some relays are used to allow a small current to control a larger one. Like if you want to turn neons on with your head unit's remote. You can't wire them directly, because the HU's remote circuit would burn out. So you use the HU to drive a relay's coil, and the relay contacts become your switch for the neons.
Relays are also used for isolation, because there's no electrical connection between the coil and contacts. You could switch 110vac circuits with a 12vdc signal, for example.
Relays can also have multiple sets of contacts that all switch at the same time. Then you can switch a whole bunch of isolated circuits with only one control signal.
Then there's Solid State Relays... but I digress...