Regulators become a necessity when your technique or the viscosity of the liquid requires you to control airflow more precisely. Having a compressor that gives you a wide range of pressure to regulate allows you to experiment with a broader range of paints and techniques.
A pressure regulator and gauge combination controls air output to the airbrush. It's not absolutely necessary to use a regulator, but once you use one, you'll never want to work without it.
A moisture trap attaches directly to your compressor, and reduces the amount of moisture in your airline. It is particularly important in warm or humid conditions.