Why does my battery keep dying, and when I jump it clicks but doesn't turn over. on 2003 Hyundai Elantra

Automatic Transmission. Replaced battery about a month ago because it kept dying. Got out of work today and the car was dead. Hooked up cables to charge. All the lights came on. Turned key and heard rapid clicking but didn't start. Waited a bit longer but when i turn the key no clicking. The lights all stayed on in the car and on dash board.

Asked by for the 2003 Hyundai Elantra
danny, you could have a battery drain somewhere (since you did not mention a battery light, i am assuming the alternator is good) but check the alternator and battery first. The rapid clicking is the starter solenoid trying to work with a low charge. possible that the battery is totally drained again.
I just found out the previous owner had installed a remote starter.