I have a 1998 Chevrolet Cavalier with 236000 miles. I have only had it on the road since April 2008. On three occasions the car has been dead when trying to start. I mean turn the key and nothing happens. No lights and no engine trying to start sounds. On one occasion a jump start got it going. On another occasion a black box under the glove compartment was clicking and I disconnected the wire harness and hooked it back and the car started when I tried it again (battery was not dead). This is the third time same symptoms. I disconnected and removed the box (Google says it is a keyless alarm module and I also disconnected the horn relay. A jump start then started the car and I let it run for an hour and then cut it off with parts still removed. Next morning the car was dead again. Took the battery out to charge and charger acts like the battery is not discharged. Any ideas what could be wrong?

Is the alarm system from the factory or aftermarket? It could very well be malfunctioning and telling the car that it's being stolen, which could result in it not starting. It could also just be the starter.

I researched the piece using the GM part number and got a Google response that it was a Keyless alarm module. When it is removed / disconnected the dome light doesn't work and the seat belt and door open with key in alarms don't go off.

I don't think it is the starter because the entire electrical system appeared to be dead. When the car wouldn't start the box was clicking like a relay chattering. I just trickle charged the battery for several hours and reassembled everything and it started again.

The chattering-clicking is from the battery being dead. It could be just that the battery is bad but there could also be a draw somewhere that's causing it to keep draining down.