Representation theory for manifolds
A topological quantum field theory, despite his frigthening name, is simply the idea to associate vector spaces to manifolds of dimension $n$ and linear maps to manifolds of dimension $n+1$ with boundary. The linear maps are between the vector spaces associated to the boundary. Gluing together manifolds results in composing the linear maps. TQFTs were introduced by combining basic principles of quantum mechanics with general covariance.
As for so many other concepts, John Baez blog helps a lot: