In the geometry of manifolds, a manifold M given abstractly is considered as a candidate to be embedded in Euclidean space of given dimension n (at least dim M, naturally: see invariance of domain). That means we look for a submanifold of n-dimensional Euclidean space that is at least homeomorphic to M. If M is smooth we shall want a diffeomorphism; in the case of a Riemannian manifold an isometry (cf. Nash embedding theorem). The interest here is in how large n must be, in terms of the dimension m of M. The basic results of differential topology here concern the case n = 2m. For example the real projective plane of dimension 2 requires n = 4 for an embedding. The less restrictive condition of immersion applies to the Boy's surface - which has self-intersections.
In fact the occasion of the proof by Hassler Whitney of the embedding theorem for smooth manifolds is said (rather surprisingly) to have been the first complete exposition of the manifold concept (which had been implicit in Riemann's work, Lie group theory, and general relativity for many years); building on Hermann Weyl's book The Idea of a Riemann surface.
Based on an article from FOLDOC, used by permission.