If you are having problems with your battery, here are a few quick checks to perform,
1. You will need a digital multi meter or dmm, to do these tests.(you should have one anyway)
2. First you need to check all your cables and connections on the positive and negative sides. Make sure the cables are clean and tight.
3. Then test the battery voltage, on a fully charged battery it should be 12.66 volts, if it is much lower, the battery needs to be charged first.
4. After the battery is charged, it should be allowed to settle down for an hour or so, then perform the test.
5. Then you need to check the voltage on the battery while the starter is engaged, this is a load test of the battery with the starter and engine as the load. A good battery should not go below 10 volts on this test. If it is lower than that, the battery is going to fail on some cold morning.
The battery can have a shorted cell also, which will reduce voltage to 10.55 volts, this can happen at any time. It happened to me once when I was getting gas, after refueling, I went to start my car and nothing happened, because that is not enough voltage to engage the solenoid. And it is no use to try and jump start it because it will not work. You need a new battery right now!
You can also have a problem with an open battery, where you won't get any voltage, again the only solution is to replace the battery.
Once the car starts, the alternator becomes the source of power instead of the battery, the battery then becomes a load.
I have heard some people say to disconnect one of the battery cables to check if the alternator is charging, but I think this is not a good idea. Because the alternator will sense low battery voltage and will try to charge it at a very high output, which causes spikes, which can damage the diodes in the alternator. If you have a battery with low voltage it is not a good idea to use the alternator as a battery charger either, because the heat from working to recharge a dead battery can also damage the diodes.