a smooth manifold
Imagine you are an ant on the surface of a huge ball. From where you stand the ground looks like a flat patch — you could draw a little map of your neighbourhood on a sheet of paper, with x and y coordinates, and it would describe your immediate surroundings faithfully. You cannot do this for the whole sphere at once with one flat map (any world map distorts), but you can cover the sphere with many small overlapping maps, each flat and ordinary. A smooth manifold is exactly the abstract version of this idea: a space that is not flat globally, yet looks like ordinary flat space (R^n) in a small region around every point.
Precisely: an n-dimensional manifold is a space M such that every point has a neighbourhood that can be matched, by a smooth invertible correspondence, with an open piece of R^n. Each such correspondence is a chart (a local coordinate system); a collection of charts covering all of M is an atlas, just like an atlas of the Earth. The crucial extra condition that makes it 'smooth' is about the overlaps: where two charts cover the same region, the formula that translates one set of coordinates into the other must be infinitely differentiable. That single requirement is what lets us do calculus — take derivatives, define velocities, integrate — on a space that has no flat ambient room to sit in.
The deep point, due to Riemann, is that geometry need not live inside a bigger flat space. A sphere is usually pictured sitting in R^3, but the manifold idea lets us study it intrinsically, using only quantities an inhabitant could measure, with no outside vantage point. This is what made Einstein's curved spacetime possible: a four-dimensional manifold with no 'outside'. A caution: 'smooth' refers only to how the coordinate patches fit together, not to the absence of geometric bending — a sharply curved sphere is a perfectly smooth manifold.
The 2-sphere needs at least two charts: stereographic projection from the north pole maps everything except the north pole to a flat plane, and projection from the south pole covers the rest. On the overlap (the sphere minus its two poles) the change-of-coordinate formula is z -> 1/z, which is smooth — so the two charts form a smooth atlas, and the sphere is a 2-dimensional smooth manifold.
One flat map cannot cover a sphere, but two overlapping flat maps can — and their smooth overlap is what makes it a manifold.
A common confusion: being a smooth manifold says nothing about being curved or flat. Curvature is extra structure added by a metric; the bare manifold knows only its dimension and which functions count as smooth, not distances or angles.