90% of the time you would use one say, to use an existing circuit like the Headlight switch for example, to turn on a relay instead of the new device itself.. turning on a relay requires very little power and makes little impact on the existing circuit.

The relay itself then turns on the new device, say a set of fog lamps.. you wouldn't want to run those hardwired into the headlights because it would either cause them all to run dim, or blow the fuse depending on how the factory has them wired. So you use a relay

That's one application, another would be to stop power from backfeeding to the source when you're trying to turn something on, say for example some older GM cars and the door locks, you want to put in a keyless entry so you find the wires that are normally ground but go to + when you hit the locks.. but if you put + to the wire yourself, it gets hot and either pops the fuse or worse, it's a dead short. The solution for that kind of circuit is to use a relay in line in a manor where it can "break" the circuit , when you hit the keyless entry button, the relay is tripped and it only sends the + to the actual lock motor, and not to the switch where it will short out.

So either you're isolating the power from going a certain path, or you're isolating the current draw from the factory circuit.

There's other ways too but this is kinda off topic, what I was getting at here is the circuit he's got the gauges wired into might not be happy with the way it's hooked up.. it doesn't seem like that should kill the car, but since it started happening at the same time why now give a shot at disconnecting the gauges from the headlight circuit and see if the problem goes away

 

Superb +1