When I pulled my rear wheel off to get the tire patched, I noticed that there are two lines running to the rear drums. A wimpy *** brake line and another much larger line. What does that second line do?

Hydraulic brake line and emergency brake cable?

It's not a cable, its a rubber tube and it's on both brakes, but I'm not sure where it goes.

It's the parking brake cable. It's not a hydraulic line.

I just went and looked and the driver side one does appear to be the parking break, but the passenger side is the side I'm really concerned with. The tube/cable/whatever in question going to the passenger side drum is resting on the exhaust tube and appears to be leaking. I originally thought that was the brake line and thought that's why the passenger rear brake made an odd noise whenever the brakes are applied (its like a squish...squish...squish sound, not a squeal) at low speed (unless I just can't hear it at a higher speed) until I had to get that tire patched and I noticed the dinky line hidden on top of the rear end. I took some pictures:

Yeah it's supposed to be in a little wire clip sort of like a "S" hook that hooks to a bracket up under there to keep that from happening, but it's not clipped; so the parking brake cable is laying on the exhaust and its plastic coating is melted. Bummer.

Hook it back up to the "S" hook. If the hook is gone, go find one at the junkyard. Lots and lots of cars have the same kind of one.

Nothing is "leaking", just the molten plastic is dripping down the exhaust pipe.

The brake hydraulic line doesn't have to be big for any reason; those little "wimpy" steel lines are PLENTY.

After I read your post I thought that it was just the coating that had melted and not something leaking, especially since nothing else was coming out. I'll have to look and see if there's some kind of hook up there. I know there used to be something holding that up because it wasn't always sitting on the exhaust.