Why Some Surfaces Do Not Unroll

A frequently asked question: The goal of many designers is to obtain flattened, two-dimensional patterns from their three-dimensional designs. The three-dimensional surfaces are analyzed using Gaussian curvature analysis and classified into two types of surfaces: developable and non-developable.

Developable surfaces can be flattened using exact solutions. These are surfaces like cones, truncated cones, and cylinders. ...

http://en.wiki.mcneel.com/content/upload/files/developable_PG.pdf

Some useful tips and third-party products for unrolling non-developable surfaces:

Unrolling compound curved surfaces

Applying 2d patterns to 3d developable surfaces