Think of it as an AC generator without the permanent magnet. In an AC generator, the moving coil is connected to the brushes such that the same end of the coil is always contacting the same brush. Since the current flow through the coil reverses with each half-rotation, the result is an AC current at a frequency equal to the generator's RPM.
Now take away the permanent magnet and replace it with an electromagnet. This is the field coil. When DC is applied to the field coil, you have the magnetic field needed to generate the AC power.
The main advantage of an alternator over a generator is that it produces more power at low RPMs, though I'm not sure why. The biggest drawback is that it will not generate any power at all unless there is power in the field coil, so hand-starting with a dead battery will not charge the battery once the engine's running- you need an electrical source to get it going.