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I am not an electrician or an electrical engineer and I didn't sleep at a Holiday Inn Express last night, so someone else may be able to explain it better, but a relay is basically an on/off switch that has two sets of connections. One side is the control side (lower voltage or wattage) from the operating switch and the second side operates the equipment (higher voltage or wattage). Relays can be normally open or normally closed. When the lower voltage or wattage is applied acrossed the one set of contacts it actuates the contacts on the higher voltage or wattage side and either makes or breaks the circuit, turning the equipment off or on. Using a relay avoids running the higher voltage or wattage necessary to operate the equipment through the control switch. For example on the A/C compressor. You wouldn't want to run the required wattage that the compressor draws through the switch on the dash. So they wire a low wattage control circuit through the dash switch and it pulls in ( or actuates) the higher wattage side of the relay which completes the circuit to the compressor to run the A/C.
Here's a link to an article that shows some diagrams and pictures of relays and explains some applications. It's several pages so you need to click on "Next Page" at the bottom of each screen.
Relays allow a smaller amp. switch ie; like one found on a micro chip, to switch an item with a much higher amp. requirement such as fan motors or allows one switch to do many things by first switching a relay. It's amps.mostly in cars not voltage.