A relay is basically an electromagnetic switch. A basic relay will have two circuits: low amperage (to horn button) and high amperage (to horn). The low amperage circuit in the relay is an electromagnet. When you hit the horn button the electromagnet moves a piece of ferrous metal, in the relay, so that it bridges the contacts of the high amperage circuit and the horn blows.
If the high amp contacts in the relay are burned or worn, the relay will still click (the clicking is just the sound of the ferrous bar in the relay hitting the high amp contacts) but the high amp circuit will not be complete.
Do what shady said. Or test for voltage at the horn when the horn button is pressed. If the voltage is good (around 12V) the horn may be pooched.