I have tinkered with relays while building small projects with microcontrollers in the past. Usually I use them to switch higher amperage circuits off or on. I recently purchased this horn and this wiring kit for it, along with a small 12v battery. I also had to purchase an automotive relay that was not included in the wiring kit. However, I have no idea what the purpose of the relay is, since the horn is activated using the push button supplied with the kit. Could I have just wired the + terminal from the compressor straight to the battery, then wired the - terminal to the switch, then to the - terminal of the battery?
The current drawn by the compressor will likely 'fry' (over heat) the horn button. The relay can reliably provide LOTS (30 or more amperes) of current. Also the amount of current the compressor will need for the first second to so will be VERY high until the motor is running. So the relay is quite necessary.
The horn button will work fine to run the compressor, but only for a few operations, then it will be ruined.
Note also that the relay allows you to switch the positive wire of the compressor with a horn button that is connected to ground. In effect it is inverting the polarity of the switch.
Remember that some car horn buttons will be using a small button on the indicator stalk or on a flexible circuit or brush between the steering wheel and column. These won't have a high current switching capability.