h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

The right hand screw rule has two applications (that I can think of!).

The first concerns vector geometry: given two vectors (directions with a given length), there is a third vector called the cross product, which is perpendicular (at right-angles) to both of the original vectors. But with just this information, the cross product could point in either of two directions - so which is it? This depends on the order you take the first two vectors - X x Y is opposite to Y x X.
The direction for X x Y is given by the right hand rule thus: hold your right hand straight, pointing in the X direction, palm on the left with your thumb at right angles to your fingers. Then curl your fingers until they point in the Y direction. This should only be possible in one direction (unless you are double jointed) - the direction of your thumb gives the direction of X x Y.

The second use is to determine the direction of a magnetic field, induced by an electric current. The use of the rule is slightly different depending on the setup - if the current occurs in a straight wire, imagine your right hand grasping the wire, with your thumb sticking out in the direction of current flow. Your fingers will then be pointing in the direction of the induced field.
If the current occurs in a coil, half close your hand into a fist (again, with your thumb sticking out). Your fingers represent the coil, pointing in the direction of current flow. Your thumb will then (magically!) point in the direcion of teh magnetic field inside the coil.