It can be immersed in three-space, but an embedding of it requires four dimensions, as it must pass through itself without creating a hole.  
It provides the only exception to the Heawood conjecture, as 6 colors are required to legally color its regions when the Franklin Graph is embedded on it.  Its topology is equivalent to two cross-caps with coinciding boundaries, and one can cut it and produce either one or two Mobius strips.  It is often thought of as the gluing of opposite ends of a rectangle while giving one pair a half-twist.  It can be immersed in three-space, but an embedding of it requires four dimensions, as it must pass through itself without creating a hole.  FTP, name this nonorientable, one-sided closed surface named for a German mathematician.
It provides the only exception to the Heawood conjecture, as 6 colors are required to legally color its regions when the Franklin Graph is embedded on it.  
Klein bottle (prompt on Franklin Graph before it is read)