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A smooth manifold is a topological manifold equipped with an atlas whose transition maps are infinitely differentiable. This additional structure lets you do calculus on the manifold, not just topology. The smoothness condition on overlapping charts ensures that the notion of "differentiable function" is well-defined regardless of which chart you use.
You already know from topology that a topological manifold is a Hausdorff, second-countable space that is locally homeomorphic to ℝⁿ. An atlas is a collection of charts (homeomorphisms from open sets of M to open sets of ℝⁿ) that covers the manifold. On a topological manifold, you can talk about continuous functions, but not about derivatives — the concept of differentiability is not invariant under arbitrary homeomorphisms. A smooth manifold adds exactly the structure needed to make differentiation well-defined.
The key idea is transition maps. When two charts (U, φ) and (V, ψ) overlap, the composition ψ ∘ φ⁻¹ maps one coordinate patch to another, and this map goes between open subsets of ℝⁿ where we already know what "differentiable" means. A smooth atlas is one where every transition map is C∞ (infinitely differentiable). Two smooth atlases are compatible if their union is again a smooth atlas. A maximal smooth atlas — a smooth structure — is one that contains every compatible chart. In practice, you specify a small atlas and note that it extends uniquely to a maximal one.
With a smooth structure in hand, you can define smooth functions f : M → ℝ (those whose coordinate representations are smooth), smooth maps F : M → N between manifolds, and diffeomorphisms (smooth bijections with smooth inverses). Diffeomorphism is the appropriate notion of "sameness" for smooth manifolds — just as homeomorphism is for topological spaces. The inverse function theorem from multivariable calculus transfers directly: a smooth map whose derivative is invertible at a point is a local diffeomorphism near that point.
Common examples of smooth manifolds include ℝⁿ itself (with the identity chart), the sphere Sⁿ (with stereographic projection charts), the torus T² (with angle-based charts), and Lie groups like GL(n, ℝ). The product of smooth manifolds is a smooth manifold. Open subsets of smooth manifolds inherit smooth structures. Level sets of smooth functions are smooth manifolds when the derivative has full rank (by the implicit function theorem) — this is how most concrete manifolds arise in practice.
The smooth manifold concept is the foundation for everything in differential geometry. Tangent vectors, vector fields, differential forms, Riemannian metrics, connections, and curvature are all defined using the smooth structure. Without it, you have topology but not geometry. The entire apparatus of differential geometry rests on the ability to differentiate, and the smooth atlas is what makes differentiation coherent across an entire manifold.