(On preview, I see Rick has given an excellent account, but I’ll chip in my two bits with a slightly more detailed description of a quick and dirty alternator test.)
There is a simple way to get a good idea if your alternator is charging properly. It won’t tell you everything, but if the results are good the alternator is probably ok, and if it fails you know the alternator (or regulator) is definitely bad.
You just need a simple volt meter. Set it to read voltage for DC voltage, in the lowest range that will read at least 18 volts.
With the car not running, put the positive lead from the voltmeter on the positive terminal of the battery, and negative lead on the negative terminal. You should read about 13 volts or a little under.
Start the engine, and check the voltage again. If the alternator and charging circuit are working correctly, you should see the voltage move up to somewhere in the range of about 14 to 15 volts. (I think the official range for most cars is 13.7 to 15 volts). For example, my dash gauge sits just a little under 14.5 volts pretty much all time.
If your alternator reads in this range, it should charge the battery without any problem, keeping in mind it may take a few minutes.
If the voltage is below that range, the battery won’t charge correctly. If it’s higher, the battery can be overcharged and damaged permanently, and even explode!
In either case, the alternator, regulator, or something else in the charging circuit needs to be looked at. It could be something simple like a loose belt, or many other things.
Good luck!