What he is saying is that a car gets the power to START from the battery. Once the car is RUNNING, the alternator is spinning and supplying the power to the car and to recharge the battery.
Generally you can take voltage reading at the battery with the car off and should get ~12V, then when the car is running you should get something more like 14V (the charging voltage of the battery). If the car is running and still showing 12V or less, then likely the alternator isn't any good. You can pull off an alternator and take it to an auto-parts store for bench-testing.
So, new battery and it started up? But, now it won't start again? That may suggest that it is the alternator. Also, double check that all the cable connections are tight.