A differential manifold is a mathematical structure that generalizes the concept of curves and surfaces to higher dimensions. It consists of a set of points that locally resemble Euclidean space, allowing for the application of calculus. This means that, around each point, you can define concepts like tangent vectors and differentiable functions. Differential manifolds are essential in various fields, including physics, where they provide the framework for theories like general relativity. They also play a crucial role in geometry and topology, helping mathematicians understand complex shapes and spaces through smooth transitions and continuous transformations.