The alternator should have pretty much a direct connection to the battery, so if you measure the voltage at the battery and it isn't at least 13.0 volts when the engine is running at a fast idle then the alternator isn't working.
If you have an external voltage regulator (very unusual these days) then the problem could be the regulator. The more common alternator these days has only one connection (well, two if you count the frame of the alternator, which provides ground). External regulator or internal the main power is transfered through a large wire bolted to the alternator.
While it is possible, and even common these days for there to be only one wire to the alternator it is possible that there could be additional wires. Alternators need a little power to generate power. It used to be common that this initial power was supplied to the alternator from the ignitions circuit via the "idiot light", the light on the dash that was typically labled "Alt". If this light burned out then the alternator wouldn't get its initial excitation voltage and might not work.
If the alternator has an external regulator then there will certainly be additional wire(s) on the alternator, and several wires connected to the regulator. The regulator could have failed, or one of the wires into/out of the regulator could have broken.
The small wire of importance on the alternator would be the "field" wire, which would typically be labeled "F". With a typical external regulator I would expect this terminal to be 0 volts when the ignition was off, ~12 volts when the ignition was on but engine not running, and some lower voltage when the engine was running and full output of the alternator was not needed.
If you can identify this connection measure the voltage on it. If you don't see any change between ignition on or off then I would replace the regulator. If you see it go from zero to 12 volts then I would suspect the alternator.
You can force the issue by using a jumper wire to connect 12 volts to the field terminal. With the engine running ~ 1200 RPM you should see the battery voltage step up dramatically (from 12 volts to 13 or so) when you touch 12 volts to the field terminal.
Likewise, if your alternator has an "idiot light" setup that connects directly to the alternator then there should be a small wire that has 0 volts on it with the ignition off and some small voltage on it when the ignition is on.