Smooth Manifold
Definition. A topological space $M$ covered by charts — homeomorphisms $\varphi_\alpha: U_\alpha \to \mathbb{R}^n$ — such that every chart-overlap map $\varphi_\beta \circ \varphi_\alpha^{-1}$ is smooth. The collection of charts is an atlas.
Intuition. A space that locally looks like $\mathbb{R}^n$ but may be globally curved or twisted ($S^2$, torus). No embedding in a bigger space is assumed — all structure comes from the charts and their compatibility.
Physics anchor. Spacetime. In Landau Vol. 2, Minkowski space needs one global chart, which is why the manifold machinery stays invisible. In GR (and around monopoles) it stops being invisible.
Key point for bundles. The pattern local model + smooth gluing on overlaps is reused verbatim to define a Fiber Bundle: replace "looks like $\mathbb{R}^n$" with "looks like $U \times F$" and chart-overlaps with Transition Functions.
Related: Tangent Space · Vector Field · Pushforward and Pullback · Module 01 - Smooth Manifolds Refresher
Sources: Sternberg Ch. II; Landau §1–§7 (implicitly).