a smooth structure
Vol I gave you a topological manifold: a space that looks locally like R^n through charts, with continuous transition maps where charts overlap. But 'continuous' is too weak to do calculus — you cannot differentiate a merely continuous function. A smooth structure is the extra layer of bookkeeping that lets you say which functions on M are differentiable, and to do so unambiguously no matter which chart you happen to be standing in.
Concretely, a smooth structure is a maximal atlas all of whose transition maps are C^infinity (infinitely differentiable) as maps between open sets of R^n. 'Maximal' means we have thrown in every chart that is smoothly compatible with the ones we started with, so the structure does not depend on an arbitrary choice of starting atlas. Once it is fixed, a function f: M -> R is called smooth if its expression in coordinates, f composed with each chart's inverse, is an ordinary smooth function on R^n; the chain rule guarantees this verdict is the same in every overlapping chart.
Two honesty points that matter at this level. First, a smooth structure is genuinely extra data: a topological manifold need not admit any smooth structure (such manifolds exist in dimension 4 and up), and when it does, the structure need not be unique. The 7-sphere carries 28 distinct smooth structures (Milnor's exotic spheres), and R^4 carries uncountably many. So the bare word 'manifold' never by itself means 'smooth' — always say which you mean. Second, replacing C^infinity by C^k or by real-analytic gives a different and weaker or stronger notion; the C^infinity choice is the standard one because it is closed under all the operations of calculus while staying flexible enough to admit partitions of unity.
On the circle S^1, two charts — stereographic projection from the north pole and from the south pole — overlap on S^1 minus two points, and there the transition map is t -> 1/t, a smooth function on R minus 0. So these two charts are smoothly compatible and generate the standard smooth structure on S^1.
Compatible charts whose transition is smooth define one smooth structure.
Do not confuse the topological manifold with its smooth structure: the same underlying space can carry inequivalent smooth structures, and some carry none. 'Smooth manifold' always means a pair (space, chosen smooth structure).