Difference between open and closed Universes?

[Mind Bender]

What is the difference between a closed Universe and an open Universe? Please explain in layman's terms and describe what type our Universe is.

[bapowell]

Geometrically, an open universe has a negative Gaussian curvature, K = -1. One way to visualize this geometry is as the 2D surface of a saddle in 3D space. A closed universe has positive Gaussian curvature, K = +1. A closed universe is readily visualized as a sphere. It is not currently known what the global geometry of the universe is; however, the observable universe is very close to flat, K = 0.

Einstein's General Relativity shows us that the geometry of space is determined by its energy/matter content, and so each of the above geometries also has a characteristic energy density: closed universes are overdense, open are underdense, and flat are at critical density -- they're "just right".

Wikipedia has a good discussion of these ideas: http://en.wikipedia.org/wiki/Shape_of_the_Universe

I'd recommend looking there for general questions like this first, then come to PF with more specific questions.