Hyperboloid of One Sheet
The hyperboloid of one sheet is a surface of revolution obtained by rotating a hyperbola about the perpendicular bisector to the line between the foci (Hilbert and Cohn-Vossen 1991, p. 11)
Another construction:
Given are two congruent circles in parallel planes with centers on a line perpendicular to the planes. Connect every point of the bottom circle with the point directly above it on the top circle. These lines will form a cylinder. Now, keep the bottom circle fixed and rotate the top circle. The rotated lines will form a hyperboloid of one sheet. For rotation on angle the surface is a cone.
The applet below shows that both constructions define the same surfaces.