As Jesper noted, the method to call is repaint which calls update which eventually calls paint. The manner in which the JVM provides the Graphics reference is not all clear to me, and then there is the thing called a 'graphics context'. Well, I suspect the graphics context is just a way of saying that (my words) 'the paint method being passed a reference of type graphics, provides a context in which to direct the graphical display of a component (of which method paint is a member). This looks like a good example of delegation. The component delegates all of its graphical manipulation to some hidden object accessed through the reference passed to the method void paint(Graphics g).
The reference of type Graphics allows access to all of the abstract methods in the Graphics class. (Java2 API doc is worth a peak). The methods manage the graphical state of the component.
To use the Graphics methods, place calls to them from a components' paint method, but you don't call paint directly. As Jesper noted, invoke the method repaint and it will in turn call update which eventually calls paint and runs the Graphics methods invoked within paint's body.