Hyperbola
A Hyperbola is obtained when a section plane, parallel/inclined to the axis cuts the cone on one side of the axis. This is illustrated in figure 1. 
A Rectangular Hyperbola is obtained when a section, parallel to the axis cuts the cone on one side of the axis.

 

Figure 1.  illustration of a hyperbola.

Hyperbola (Applications)

Hyperbolic shapes finds large number of industrial applications like the shape of cooling towers, mirrors used for long distance telescopes, etc. (as shown in figure 2)

 

Figure 2.  use of hyperbolic shapes in engineering applications. (source: internet)