Let me think about this.Alright then, Electrodynamics is surely the easiest example.
Take R^4 as a manifold and assign an additional degree of freedom to each point*. In the EM case, that is an element of U(1), think of some ring everywhere in spacetime. There's now a connection on this bundle, and the connection coefficients turn out to be 1-forms that describe the electromagnetic potential. The associated curvature is F=dA, the electromagnetic field strength.
*: I guess that's the fiber, but I haven't found it stated like that explicitly