Horn won't work

What model/ year vehicle?

Horns are pretty simple. You already know the horn works, "direct wired."

The relay has three connections. One obvious large wire should be hot at all times.

The horn hooks to a second

The third goes to the horn button, and is grounded to key the relay.

You should be able to work the horn ring/ button and hear the relay click. If not, make sure you have battery to the relay, go out and short the button wire with a clip lead to ground, and if that works, then you have a wiring issue to the button, or in the button itself.

Obviously, if the relay won't work when manually shorted, it just might be the relay.

And yeah, make sure the horns are grounded. Best way to do that is install them, hook them up, and troubleshoot right on the car. You can short across the battery/ horn terminals on the relay and they should work. Watch yourself, you can get a nasty "inductive shock" that way. They work just like an ignition coil.