One of the most profound ideas in modern mathematics. A manifold is a space on which you can do differential geometry. Roughly, it is anything which is locally like flat Euclidean space. For instance, the surface of a sphere is a two-dimensional manifold, because to a bug crawling on the sphere (or a person on the Earth) the surface nearby looks flat. Likewise spacetime is a manifold, of four (or maybe ten or more) dimensions.