A relay is like a high current remote switch. You supply 12V fused power to the relay. This is generally straight off the battery with heavy gauge wire (10-12 gauge depending on your load and relay capacity). The power for your headlights will then come from your battery through the relay to your lights. The relay is switched on/off with a "signal" wire (generally your stock headlight wire). This signal has a very low amp draw compared to your headlights.
So you benefit from full power to your headlights (13.8V in a running vehicle) not the 10-12 you may be getting now (you lose some of the power as it travels through your wiring harness, switch, and connectors). It also should cause less stress to your headlight switch (mainly if you have high power lights).
Oh yeah, don't forget to upgrade the ground too.