Please post the model or a picture of your system. It really depends on what kind of alarm system you have. For example a honeywel vista 15 or vista 20, has a 16.5V transformer plugged into the wall, then there is a box that contains the brain of the system that usually goes in a closet or basement. this box contains the brain of the system. The main board in this box converts the 16.5v AC (actually in the neighborhood of 14V) to 12v DC current. The vista system will have some leads coming off of the board just above the first two terminals on the left (these are the terminals where the 16.5 AC power is). They are a black and a red wire with a small metal "plug" on the end. These attach to a lead acid battery that is usually located in the lower right side of the box.
The vista alarm is pretty common, but yours may differ. Like some of the other posters said, most alarms are setup to take a battery backup, you probably just need to replace the existing battery.