You may want to check the rest of the camera functions if you are sure the film advanced but still don't get any pictures.
The AE-1 is an aperture priority design camera so you select the shutter speed and the camera automatically adjusts the aperture.
Sometimes the aperture control system fails to work correctly either because the lenses iris (aperture) fails to close as its gummed up or the magnet the stops the iris at the correct setting failed causing it to fully close.
Looking though the camera with the door open manually, using the Bulb setting, set the aperture to the smaller F stop (i.e. 22) see if the iris closes and opens as you adjust it. If it can't close or is slow to close you'll need to get the lens serviced as the iris blades have gotten oiled up.
The other test here is to setup a flash light aimed at you in a darkened room. You'll need to have a quick eye as you want to look though the back again and using a slower shutter speed 15th of a sec. try to get an exposure where the lens needs to close the iris just slightly. Now look though the back as you fire the camera you should be able to see the iris staying open if it shuts down fully the aperture control system is failing to stop the iris from closing.