Some bad information in this thread. Don't know where to begin but here goes:
The battery is not used JUST to crank the engine over, in fact it would be safer to say that the alternator is used to maintain power keeping the battery where it needs to be. The alternator shouldn't be used to recharge the battery once it's been drawn down either. It can be done but it shouldn't be.
If you've got 12.63 volts at key off then it's only expected to read the same voltage at the alternator with the key off. With the engine running the voltage should be at least 13.6 or higher on a good battery, 14+ volts if you're recharging a worn down battery.
If the voltage you measured with the engine off is the same as it is when the engine is running then your alternator may not be charging. Start by checking the voltage on the primary wire at the alternator, it should be the same as what you read at the battery. Next, do a voltage drop test from that wire to the positive terminal on the battery. This is done by putting the meter in DC Voltmeter mode and connecting one lead to the alternator post and one to the battery positive terminal. If you read a voltage higher than say 0.1 or 0.2 volts then there's high resistance in the wire.
The next thing would be a voltage drop between battery negative and chassis ground again, you'll want a low voltage like the last test. Also, do a voltage drop between battery negative and engine ground. Both those tests will show if you have a bad connection between the chassis and the battery AND then a bad connection between the battery and engine. Correct which ever has the highest voltage reading.
Next you'll want to check for a current draw. Put the meter in to amp meter mode, this usually requires moving the red meter lead and switching the setting to amps (cheaper meters will measure up to 10 amps while higher end meters will show 20+ amp test capability). Disconnect the negative terminal, make sure the key is out of the ignition and door is shut. Connect one lead to the cable end and the other end to the battery post. The amount that is displayed on your meter is how much of a current draw you have. As an example a headlamp will be somewhere around 3 amps, an interior light will be 0.3-0.6 amps. On a newer car (10 years or newer) you may have to wait up to 30 minutes for modules to go to sleep. For an early OBDII car or older you can usually wait about a minute or less for any draw to go away.
If you're using a Wal-Mart battery then all that testing is not needed, just take the NEVERSTART battery back and get a real one.