A differentiable 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 definition of calculus concepts like derivatives. This means that, around each point, you can find a neighborhood that behaves like flat space, making it possible to perform calculus on the manifold. Differentiable manifolds are essential in various fields, including physics, where they are used to describe the shape of spacetime in general relativity. They also play a crucial role in differential geometry and topology, providing a framework for understanding complex shapes and their properties.