The horn on my Chevy constantly is going off.

The clicking noise is a relay tripping on and off, if you can isolate which relay it is that might point you in the right direction.

What's a relay switch I don't know much about cars.

A relay controls other devices (horns,starters,etc.) through the use of a switch. If you still have the owners manual it should have a description or legend of the what's in the fuse box as to what fuse or relay goes to what.

The ignition 1 that clicks the horn stops when we take the fuse out..

Possible body control module issue, it controls the security system.

Would you happen to know how much that usually costs?

Not off hand and besides you might want have it checked out to make sure that's what it is.