The one-plane swing is more of an around-the-body swing, and the wagon-wheel of the swing plane tends to be flatter. Imagine taking the club along the curvature of the wheel and bringing it down to the ball on the same line, then following through along the curvature of the wheel. Everything happens along a single plane. For some golfers, it helps to think of this action as a baseball swing aimed at a ball on the ground. The best example of a one-plane swinger is Ben Hogan, but Tiger Woods has moved more toward a one-plane swing as his golf game has matured. As a young professional, Woods had more of a two-plane swing.
The two-plane swing is more complicated. Most two-plane golfers take the club back above the curvature of the wagon wheel, then drop the club under the plane of the wheel on the forward swing. Some golfers will follow-through above the plane, as well, which creates the famous "reverse C" finish that was popular in the 1970s. Essentially, the first plane of the two-plane swing--the backswing--is steeper than the forward swing, which has to be shallower and more "on plane" to strike the ball firmly. So the arms work on the more upright plane, while the body rotates around another flatter plane. Jack Nicklaus has the most famous two-plane swing in history, but for a vivid example of the upright take-back and the flatter downswing, check out Jim Furyk.
A one-plane golfer's swing will get stepper as the club gets shorter. The driver plane will be flatter, for example, than a wedge swing, which will be much steeper because of the length of the club. But the club still will travel along the same plane going back and through. Two-plane golfers need to be careful they drop back on-plane during the downswing or they can come "over the top" of the swing plane and hit slices or severe pull-hooks. Some players mix one-plane and two-plane swings; they swing more around their bodies with the driver and lift the club on an upright plane with their irons. They do this because they can sweep the ball off a tee with the driver, but they must hit down crisply on the ball with their irons.