Best way to check is with a volt meter. Get a reading off the battery with the car off. A full battery should read a little over 12V. Start the car and test again; if the alternator is charging the system, you should get a reading of at least 14V. Any less, and you aren't getting full performance. Less than 12V and you're not charging at all.

If the alternator is NOT charging, you can probably save the price of an entire alternator by swapping out the voltage regulator and brushes - an assembly that's inserted into the alternator and held on by a couple screws in the back of the alternator. My guess is you can get this for $75. (This will get it charging again, but it won't address other possible problems such as worn shaft bearings.)