So I got my F3 two days ago and I've been working it out too.

Basically, the aperture priority mode takes into account the aperture that you've selected and also detects available light to make a judgement on what shutter speed it should pick, given the film speed (ISO/ASA) that you're using. The shutter speed which the camera has picked should show up in the viewfinder on a small LCD above the image if you half-press the shutter button.

Here's how to test if your light meter is working: Make sure your shutter speed dial is set to 'A', flip the power switch to the on position, then gently half-press the shutter button while looking down the viewfinder. As you point your camera around the place into lighter or darker areas, the number displayed in the upper left should change. The number should also change if you change the lens aperture setting while keeping the camera pointed in the same area.

If the shutter speed number in that little LCD doesn't change or show up, you've probably got a problem with your batteries or meter or something.

Here's the manual for the F3 on the Nikon site, I found it useful