1. An n-dimensional manifold is a topological space whose points have neighborhoods homeomorphic to an n-dimensional ball. That's too technical, so I'll say the same thing without the jargon. 2. An n-dimensional manifold is such that every point has a region around it like an n-dimensional ball. Dropping the jargon makes it possible to equivocate about what "region around it" and "like ball" mean. Understanding the idea of a manifold requires at least a few examples, and the torus as a re-entrant rectangle is a good one. It is necessary to assert, though perhaps not to prove, that the neighborhoods of the edge points and the corners behave just like the neighborhoods in the middle of the rectangle. The Mobius strip provided another easily understood example, and identifying opposite points of the one edge to get a Klein bottle provides an example with enough difficulty in visualization to show that the idea is non-trivial.