Vibronic Coupling is a contraction of "vibrational" and "electronic" coupling. There are a couple of things to keep in mind. First of all, think about the molecule as a quantum harmonic oscillator. Basically, the ground and excited electronic states are approximated as parabolas (just like masses on springs).
The Born-Oppenheimer approximation says that nuclear motions are frozen on the time scales of electronic transitions (because they are orders of magnitude heavier). So electronic transitions are vertical.
The "coupling" part refers to the need for some vibrational mode of the ground state to have some degree of overlap (or similarity) with the excited state and the inverse. Ie, the transition occurs from the first vibrational level of the GS to the first, second, third.... of the excited state. The excited state relaxes to a thermally equilibrated excited state (THEXI state). Then, because of the BO approximation, the transition is again vertical--from the THEXI state to the first, second, third.... vibrational levels of the ground state.
Try to google Born Oppenheimer approx, Huang Ryhs factors, or anything about Stokes Shift.