An n-dimensional manifold is a topological space which is locally Euclidean. That is, every point lies in a region which looks like the space Rn.
A 1-manifold is a curve, a 2-manifold is a surface, etc.
Manifolds which have a structure which allow differentiation to be performed are called differentiable manifolds and if differentiation can take place arbitrarily often they are called smooth manifolds.