Not an original source, but the Wikipedia page [0] gives a good introduction. A contact manifold is then just a manifold equipped with a contact structure.

A more in-depth discussion can be found in many differential topology texts, e.g. page 581 of [1].

[0] https://en.m.wikipedia.org/wiki/Contact_geometry

[1] Lee, Introduction to Smooth Manifolds https://math.berkeley.edu/~jchaidez/materials/reu/lee_smooth...