A smooth manifold is a mathematical space that resembles Euclidean space at small scales but can have a more complex global structure. It allows for the definition of smooth functions, which are functions that can be differentiated as many times as needed. This concept is essential in fields like differential geometry and theoretical physics. Smooth manifolds can be thought of as a collection of overlapping patches, each resembling Euclidean space. These patches are connected in a way that allows for smooth transitions between them. Examples of smooth manifolds include circles, spheres, and tori, which have unique properties that make them interesting for study.