Here is a smooth surface of degree three, Clebsch's
cubic. There are exactly 27 lines on the
surface (as on every smooth cubic surface in
projective three-space).

In projective three-space with homogeneous coordinates (x:y:z:w), Clebsch's cubic is given by the equation

x^{3} + y^{3} + z^{3} +
w^{3} - (x+y+z+w)^{3} = 0

The nice computer picture shown above was made by Stephan Endrass using his program SURF. (SURF can draw pictures of a surface in three-space when you provide an equation of the surface.)

If you are interested in a more »modern« surface, don't forget to have a look at Barth's sextic.