The Airbrush tool allows paint to build up if you stay in one place for a moment. Note that the paint buildup isn't a selectable option; rather, it is the behavior of the Airbrush tool. In other aspects, the Airbrush tool acts much like the other Paint Brush tools. You can still change the various options and settings in the Tool Options window, and you can still use patterns and textures.
To see how the airbrush effect works, select the Airbrush tool (the sixth icon from the bottom in the Tool palette; it looks like a spray can) and draw a few lines, stopping every so often. When you stop, keep the mouse button down; you'll notice that the paint builds up. This effect is more noticeable with Opacity set fairly low. Even then, the brush will build up paint until it completely covers whatever is beneath it.