Test the old horn by applying power directly. If it blows, you know it's the wiring between the relay (what makes the clicking noise) and the horn itself. If it doesn't blow, you need to replace it.
Now for a very basic question: how do I apply power to the horn directly? I am not experienced in this area.
First find the horn. Second detach the main power lead going to the horn (it ground itself through the mount in most cases). Next, get a length of wire which you can attach to the positive side of your battery. The wire should be long enough to reach the horn. Apply the opposite end of the wire to the horn tab and see if noise erupts. NOTE: Do not touch this to any part of the car besides the horn tab or sparks will erupt and you could burn yourself. You only want to hold the wire on the horn tab long enough to see if there is noise. You should not get shocked from this as this is a low voltage application (if you don't believe me, put your two index fingers on the two battery posts ... nothing).
If the horn does make noise, you can use a multi-meter to test to see if there is power at the wire you just disconnected from the horn. You'd have to have someone press the horn button while you are looking at the multi-meter, but you should notice the needle swing over to 12vdc if you have it setup correctly.