In general, the point of a relay is to use a small amount of power in the electromagnet -- coming, say, from a small dashboard switch or a low-power electronic circuit -- to move an armature that is able to switch a much larger amount of power. For example, you might want the electromagnet to energize using 5 volts and 50 milliamps (250 milliwatts), while the armature can support 120V AC at 2 amps (240 watts).
Relays are quite common in home appliances where there is an electronic control turning on something like a motor or a light. They are also common in cars, where the 12V supply voltage means that just about everything needs a large amount of current. In later model cars, manufacturers have started combining relay panels into the fuse box to make maintenance easier. For example, the six gray boxes in this photo of a Ford Windstar fuse box are all relays:
In places where a large amount of power needs to be switched, relays are often cascaded. In this case, a small relay switches the power needed to drive a much larger relay, and that second relay switches the power to drive the load.
Relays can also be used to implement Boolean logic. See How Boolean Logic Works for more information.
For more on relays and related topics, check out the links below.