How does the Piper Archer III electrical system work?
I think you misunderstand how the alternator, battery, and electrical system are connected. An alternator puts out DC voltage into the electrical bus. It is used to provide electrical power to aircraft systems, as well as to recharge the battery from cranking and pre-start light and radio usage.

You can see this in the wiring diagram in the POH:


You can see the alternator about 2/3 of the way up the diagram, just to the left of center. The 'B' lead coming out of it goes directly into the ammeter and to the DC power bus (the big bar along the top).

The path back to the battery is a little more complicated, but there where it says "Battery Feeder", you can see it goes through a 70A fuse, down to where it connects to the battery contactor. So when the alternator switch is on and the battery switch is on (closing the battery contactor), the alternator can charge the battery. [In small Cessnas this is a split master switch, but I'm not sure what it looks like in an Archer.]

To answer your question specifically about what the ammeter shows, this will give you the amount of current the alternator is providing in amps. After startup you should see slightly more than usual as the battery recharges. You will probably see the number go up while you are transmitting on the radio. If you see this number get really large for no reason, there may be a fault in the electrical system somewhere. If you see this number go to ZERO, it's possible that the alternator has stopped working completely and your lights and radios are now running on battery for a limited amount of time!

I hope that is helpful. Note that nothing here should suggest a specific course of action -- this is just general electrical system information. Consult the emergency procedures or an experienced flight instructor for specific information about your airplane.