In projective three-space with homogeneous coordinates (x:y:z:w), Clebsch's cubic is given by the equation
x3 + y3 + z3 + w3 - (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.