the action

A number assigned not to a single instant but to an entire path. You take the Lagrangian, evaluate it moment by moment along a trial trajectory, and add it all up over time. Different paths receive different totals; the action is, in effect, the score of a path.

It is written S[q] = integral from t1 to t2 of L(q, q_dot, t) dt. It is a functional: its input is a whole function q(t) and its output is a single real number, with units of joule-seconds, the same as Planck's constant hbar, which is no coincidence. The physical path is the one for which S is stationary under small variations that vanish at the two endpoints.

Action is the central object linking classical and quantum physics. Hamilton's principal function, Hamilton-Jacobi theory, the action variables of integrable systems, and Feynman's path integral (which weights each path by e^(i S / hbar)) all revolve around it. That its dimension equals that of hbar is precisely why hbar sets the scale at which quantum effects become important.