An n-dimensional manifold M is a topological space, such that every point x is in a region -- an open set -- which is homeomorphic to Rn (or, equivalently, to an open sphere in Rn). In 2 dimensions, the plane, disk, sphere, torus, Moebius strip, and Klein bottle are all manifolds, but three rectangles joined along an edge are not (no region of the common edge is homeomorphic to a disk). So, locally, M looks like Euclidean Space, but the overall structure may be very different. A differential manifold gives more structure, letting you do differential geometry on the manifold.
Someone should really node examples of manifolds.