Hi Travatree,
Just a few simple pieces of info that will give you a rudimentary understanding of the basics of a car's electrical system. The battery STORES the electrical energy that is needed to start the car. The alternator CREATES electricity while the car is running (and only when it is running). It alternates current between the car's consumables (ie the things that are consuming electricity while the car is running) and also directs any excess current back to the battery - thus recharging the battery. But again, this process only occurs while the car is running. Now, batteries, like everything else in life, lose their potency over time. Batteries go through repeated cycles of charging and discharging and, eventually over time they lose their ability to take a full charge - hence why batteries in everything from watches to cars eventually have to be replaced. Your battery may or may not have had it's installation date marked on it - many batteries have little stickers on them that get removed. The removed stickers represent the year, month and day the battery was installed. But sadly many don't bother with this old-school practice and just install a battery without taking this crucial step. If you are able to somehow determine the battery and find that it is anywhere near 5 years of age or older then I'd recommend possibly replacing the battery - but not without testing its amperage rating first (as another respondent suggested). No point in changing things unless you're certain they are definitely in need of changing.
Now, here's the rub... If, in fact, your alternator isn't putting out enough current that would result in the car utilizing the battery's stored electricity to help run the car. As a result of this deficit even a good battery will eventually get depleted because it isn't being recharged (ie the alternator isn't doing its job).
I concur with the earlier poster that advised you to check not just the voltage but the AMPERAGE of the battery as well. Even if the voltage is 12 volts or above, there could be insufficient AMPERAGE to start the car. To further complicate this: If the alternator is, in fact, bad then it's not been charging your battery when the car is running. This could cause even a good battery to fail. So moral of the story is get a good battery tester - or utilize a friend's - and test both voltage and amperage of the battery. If the battery is under 5 years old then it may be worth it just to take the battery out and give it to a mechanic or a friend with a battery charger. Let them re-charge the battery and then reinstall it and then retest the battery again. But if the battery is 5 years of age or older I would simply replace the battery altogether - especially if you plan on keeping the car and or plan on using it as an every day driver. If you decide to put a new battery in - make sure to remove the old one and bring it with you to whichever auto parts store from which you plan to buy the new one. Doing this will help you avoid what is referred to as a 'core charge' as well as save you an extra trip back to the auto parts store with the old battery.
Personally, I'd pull the battery and have it charged professionally and then see if the car starts. If it doesn't then it's likely the starter.
I hope this helps.
