A relay is simply an electric switch. For example you want to turn something on that uses a lot of current when something else turns on that is a low current device. Say you have a driving light that you want to turn on that needs 10 gauge wire, but you want it to turn on when you turn on a dome light that uses 22 gauge wire. A relay has four connectors on it (or possibly five - I'll get to that in a minute). Here's a diagram
C
|
A----B
|
D
A & B is the switch part of the relay, you run a wire from the +12v wire (22 ga) on the dome light to A and run a wire from B (22 ga) to ground. Then you run a wire (10 ga) from a high-amperage 12v source (like direct from the battery - warning you want a fuse on it) to C and then another wire (10 ga) to the +12v pin on your driving light (then of course one from the ground pin on the driving light to ground). When 12v runs across A & B the switch is closed between C & D and then your driving light will be on. If no voltage is applied to A - then the switch is open between C & D and your driving light will be off. So your only using the 22 gauge wire as a trigger for the 10 gauge wire.
That is a normally open relay. You can also get a normally closed relay where if voltage is applied to A then the switch is closed between C & D, and if no voltage is applied to A then the switch is open. You can also get relays that have 5 connectors - then can be setup as a normally open or normally closed. In this case you have the equivalent of a C1 and C2. If voltage is applied to A then the switch is closed between C1 and D - no voltage to A and the switch is closed between C2 and D.
Hope this didn't confuse you!