What is a manifold?

A visual explanation and definition of manifolds are given. This includes motivations for topology, Hausdorffness and second-countability.
If you want to learn more, check out one of these (or any other basic differential geometry or topology book):
M. Spivak: "A Comprehensive Introduction to Differential Geometry"
M. Nakahara: "Geometry, Topology and Physics"
J. W. Milnor: "Topology from the differentiable viewpoint"