you'll need to find the AC transformer for your system. The "panel" you're referring to is just a remote keypad, somewhere in your house there will be a metal cabinet that houses the actual control circuitry, backup battery, and phone jack. Look in closets, and the basement near the circuit breaker panel.
There will be a rather large transformer somewhere, that supplies power to the system. Make sure it's plugged in. If it is, verify that the outlet is indeed working. Check the connections in the panel, the first two terminals on the terminal strip are for the AC power in. It could also be possible that the transformer quit, but get the preliminary stuff checked out first.
Once you get the problem resolved, and the panel powered up, connect the battery.
It's probably time to replace the battery if you don't know how old it is, especially if they just unplugged the AC and allowed the battery to completely discharge. Any local Batteries Plus or something will have that battery for a fairly reasonable price.