One way to understand why quadric surfaces have the shapes they do, is to consider their traces (cross-sections) in various planes parallel to the coordinate planes. The difficult part is putting the traces together in space to form the surface. The hyperboloid of two sheets

is a good first example because, with a little analysis, it's easy to see that certain horizontal planes will not intersect the surface at all Animation 1, and that all vertical ones will meet the surface in hyperbolas Animation 2.
The hyperboloid of one sheet Animation 3

is a little harder to visualize because, in vertical planes, the hyperbolic traces change orientation when the plane gets farther from the origin than 2 in the direction of the y-axis or 3 in the direction of the x-axis. Animation 4 shows the surface and the plane x = k for varying values of k, each trace having equation

The most difficult of the quadric surfaces for students to visualize, and certainly to draw, is the hyperbolic paraboloid. Like the hyperboloid of one sheet, the traces in certain planes are hyperbolas that change orientation after a point. As an example, I use horizontal planes intersecting the surface

See Animation 5. I constructed the frames to include the plane that produces the degenerate hyperbola (two intersecting lines) lying between the two families of hyperbolas. The animation shows this reasonably well.

To see the Graphs of Quadric Surfaces:

It must be of the form z = f(x,y):

http://www.ies.co.jp/math/java/misc/SimpleGraph3D/SimpleGraph3D.html 

For Some Parametric Curves:

http://www.butlercc.edu/mathematics/Utilities/tools_multivariable_knisley/index.htm

An amazing Parametric Curve:

Cosine Tubes:    Parametric plot of [x,y,z] = [cos(x)-2*cos(0.4*y), sin(x)-2*sin(0.4*y), y]