The easiest and best way to test any battery is with a volt-meter
The link above is for a multi-meter
. These are inexpensive and are an extremely useful tool to have. For about $15-20 you can buy a meter which will test AC/DV voltage, resistance/continuity and often diodes and/or transistors as well.
A fully-charged 12v
battery will give a reading of 12v
(or very close to 12v) on a volt-meter when it is fully-charged. 6v batteries should read 6v, 1.5v batteries should read 1.5v, etc, etc.
If the reading is lower than the power output of the battery, the battery is either not fully-charged or one or more cells may be damaged. If cells are damaged, you will never get the full 12v output from a 12v battery.
There are other ways to test batteries but they are less accurate. It's possible to use a battery charger, but you can only use this to determine whether a battery is fully-charged if the battery is known to be good
. A fully-charged battery should draw 0 amps
on a battery charger, but a damaged battery may also draw 0 amps so this is not a good test.
Basically, I'd suggest you get yourself a multi-meter. You be glad you did!
Hope this helps. Please let me know if you need further assistance.