Depends on what you mean with major... How much current can the device that turns on the relay supply?

Current drawn depends on the resistance of the coil in the relay - a typical automotive relay (aka Bosch relay) will typically pull a lot (relatively speaking) more current than some other relays (like for PCB mount)

Check out an electronics supplier and browse the relays available. Find the one with the highest coil resistance that provides the switching options you need and and can handle the current required. Current drawn is then I=U/R where U=12V and R is the resistance of the coil.

You'll probably want to add a diode (1N400x series works nicely) wired in reverse across the coil in order to snuff the inductive kickback as you turn off the relay.

Edit: but compared to the current drawn by a PC, the relay draws very little power

/MIchael