A 3-manifold is a mathematical space that locally resembles three-dimensional Euclidean space. This means that around every point in a 3-manifold, you can find a small neighborhood that looks like a piece of ordinary 3D space. Examples of 3-manifolds include the surface of a sphere and a torus, which can be visualized as a doughnut shape. In topology, the study of shapes and spaces, 3-manifolds are significant because they help mathematicians understand complex structures. The famous Poincaré Conjecture, proven by mathematician Grigori Perelman, states that any simply connected, closed 3-manifold is homeomorphic to a 3-sphere, highlighting the importance of these objects in the field.