The notion of jet space or jet bundle is a generalization of the notion of tangent spaces and tangent bundles, respectively. While a tangent vector is an equivalence class of germs of curves with order- tangency at a given point in the target, jet spaces are equivalence classes of germs of smooth maps with respect to (finite) order- tangency at some point in the target.
Jet bundles were first introduced by Charles Ehresmann.
G. Sardanashvily, Fibre bundles, jet manifolds and Lagrangian theory, Lectures for theoreticians, arXiv:0908.1886
Shihoko Ishii, Jet schemes, arc spaces and the Nash problem, arXiv:math.AG/0704.3327
D. J. Saunders, The geometry of jet bundles, London Mathematical Society Lecture Note Series 142, Cambridge Univ. Press 1989.
Arthemy Kiselev, The twelve lectures in the (non)commutative geometry of differential equations, preprint IHES M/12/13 pdf