mtMkinly would be able to explain this better than I would, and so would the docs, but...

Surface degree is similar to curve degree. The higher the degree, the smoother and tighter the surface. Surfaces generated from degree 1 curves are faceted. For example, you can create the faceted shape of a cut gemstone by revolving a degree 1 curve.

The higher the degree, the smoother and tighter the curve. The lower the degree, the more influence a CV has on the adjacent region of the curve.

Maya provides a variety of curve degrees to accommodate the import and export of objects from other software products. For example, car designers often use degree 5 curves because the extra curve precision is useful for their manufacturing software.

A curve with the default degree 3 setting works best for general modeling tasks, as it provides smooth curvature without excessive CVs. If you are in doubt about which curve degree to use, use degree 3.

Hope that helps a bit