The hyperboloid is a hyperbola rotated about a central axis.  It can be made from straight lines.  The model on the right was made using skewers according to instructions given here. It is very simple to make and nice to play with.

If you like hyperboloids and paraboloids (part of a family of surfaces called quadrics), you might want to try making this paraboloid from paper.  Below are pictures of one I made from thread.

They are used extensively in modern architecture, for example in the work of Gaudi.