There are three different tests you can do to check your charging system and battery (there are others, but I'll just talk about the big three):
Voltage with engine off
Using a multimeter, select volts-dc on the selector switch, then put the red lead to the positive and black to the negative. For a fully charged battery the reading should be in the 12.5-13.1vdc. If it is much lower than this, the battery is most likely bad.
Voltage with engine running
Using a multimeter, select volts-dc on the selector switch, then put the red lead to the positive and the black to the negative. The reading should be around 14.1-14.5vdc. If it is lower than this, observe what the voltage reads over time. If the voltage drops a little at a time and doesn't regain anything, the alternator is not charging and will need to be replaced. If the reading is over 15vdc (even up into the 16vdc range) the alternator is overcharging. Take it into the parts store and have them bench test it to see what's going on.
Cranking Amps (CA) and Cold Cranking Amps (CCA) testing
Just because the voltage looks good, doesn't mean there isn't something amis with the battery. One other test you can do is to test the CA and CCA the battery should be rated at. Using a tester like this SOLAR BA5 Cold Cranking Amps Electronic Battery Tester, you can tell what's going on inside the battery if you know what it's rated at.
Looking on the battery labeling should tell you what the CA/CCA are for the battery. Plug the battery tester with red on positive, black on negative. Input which test you want to perform (CA or CCA), then input the rated amperage for the test you've chosen. Click the test button and it will spit out what the actual amount of CA/CCA the battery has. If there is a large difference between what it's rated and what the actual is, the battery will need replaced.