It could be that the battery is dead. If you can turn something else on, like the stereo, and observe it while you try to start you can establish if it is the battery or the relay/starter. If the stereo continues to play (or a light stays on reasonably bright, or the voltmeter stays above 10 volts, etc.) then the problem is not the battery.
The red button is the circuit breaker. If the breaker had tripped you would have nothing, no click, nada. The starter relay is often mounted on the same bracket as the circuit breaker so the click can sound like it is coming from that.
The starter relay would look something like this:
http://amkproducts.com/starter_relays.asp There are a number of variations but it will have two large posts (5/16" bolts) that the heavy battery cables attach to and one or two small posts that the wire from the ignition switch connects to.
If the two big posts are close together or if the terminals attached to them hang down a bit you can do a quick test by shorting across the two with a screw driver. Be prepared, it will spark, but it will be okay. If the engine cranks and sounds proper shorting across the relay then the relay is bad. If you get nothing then the starter is probably bad.
If you are replacing the starter relay be careful as there are a number of different styles and while they may look alike they can vary in operation. If yours has only one small post then it is pretty safe, but if it has two posts the function of the other post can vary. In some cases that second post connects to ground, in other cases that second post is the bypass to the coil ballast resistor.
Rod