Your system's standby battery (or batteries) should be changed every 3-5 years, which is what most manufacturers recommend.
If you're unsure of the age or last time your system was serviced, you're going to be in the market for a replacement. Also of interest, if the panel had it's AC power transformer removed and the system left on battery power until dead, damage to the battery may have resulted (called deep-cycling). Some of the newer and better panels on the market have features to eliminate this damage, but don't count on it.
Testing a standby battery for capacity can be done, but requires a decent amount of time, some extra parts, as well as a good DMM.
Better techs carry a unit that can "load test" a battery, but truthfully, the cost and effort to check a marginal battery usually pays for the replacement.
Your system's battery, generally depending on the system, looks like a plastic wrapped brick inside your main control panel, as well as with any auxiliary power supplies. It is rated in volts and amp-hours. The replacement needs to be the same voltage rating, and at minimum, the same or higher amp-hour rating.
Once you've obtained and installed the replacement, take a couple of seconds and write the date (MM/YY) it was replaced either on the unit with a sharpie or on a piece of tape for future reference.
Image courtesy of Elk Products