Ok, here's a basic explanation of how a relay works in a 12 volt environment. The example I'll use here is wiring up a set of fog lights so that they turn on automatically when you turn on your headlights.

Making connections and safety
Let me say first of all that if you're uncomfortable with wiring your car, or if you aren't sure what you're doing, please seek the help of an expert. Improper wiring can lead to damage to electronic equipment, damage to your car's electrical system, or even fires!

It's very important to make sure you make good sound connections, I recommend soldering all electrical connections. The twist and tape method or the use crimp on connectors will eventually lead to problems. It's also VERY important that you insulate all connections.

Improperly insulated connections can cause severe electrical problems and even fires. I suggest using heat shrink tubing for all connections. In cases where you can't use heat shrink tubing and are forced to use electrical tape, please use good quality tape.

Make sure you use fuses for all electrical connections. They can prevent damage to your electrical system, damage to your equipment equipment, and they can also prevent fires.

Fuses should always be installed as close as possible to the power source. If you're going to be adding a lot of accessories to your car, it might be a good idea to add a second fuse box. Tapping power from an existing fuse box is a bad idea. This also gives you an easy place to tap power without running multiple wires through the firewall.

When running wires through the engine compartment make sure the wires are well secured, and cover them with split loom. It's available from almost any car stereo store. Any wires being run directly from the battery MUST BE FUSED AT THE BATTERY! Make sure you use a fuse holder that is waterproof and meant for outdoor use.

Relay essentials
The purpose of a relay is to use a low amperage circuit to trigger a device on a high amperage circuit. For example, if you want your headlight switch to turn on a set of fog lights, hooking the fog lights directly to the headlight wire would cause you to exceed the maximum amperage rating of your headlight switch and headlight circuit. The end result would be a burned out switch, melted wires, or possibly an electrical fire.