Sounds like the alternator is working, probably just the battery. Most parts stores will test either for free.
You can do a simple check on the alternator yourself: With the engine running turn your headlights on. When you rev the engine the lights should get a little brighter, dim a bit at idle. If this happens, the alternator is producing electricity. If you shut off the engine and the lights rapidly dim, then you can bet on the battery being the likely problem.
Like I said, most parts stores can test the alternator to make sure it is operating properly.