Yup. Alternator. Once the car is running you can disconnect the battery and guess what, it will continue to run just just fine. Remember, once started the alt is both running the car AND charging the battery.
Now, there are times when a bad cell in a battery can cause a situation that disrupts the alternator and it wont output right, but thats pretty rare.
Sounds like the battery *may* be ok since jump starting it charged it up a little and the car ran off of it. But then it drained and the alt was not doing its job.
I anything, head to a junk yard to get an alt. And as other said, get a volt meter.