What is the specific role of the cerebellum when it comes to 'coordinating movement'?
The principal function of the cerebellum, which was detected years ago, is to calibrate detailed movements rather than initiating movements or deciding which movements to execute (Ghez et al, 1985). This was concluded by observing the changes which occurred after damaging cerebellum. Animals and humans with cerebellar dysfunction show, above all, problems with motor control, on the same side of the body as the damaged part of the cerebellum. They continue to be able to generate motor activity, but it loses precision, producing erratic, uncoordinated, or incorrectly timed movements. Thus, cerebellum helps in coordinating fine-tuned movements and inhibits involuntary movements. Apart from this, fMRI studies have indicated that more than half of the cerebellum is intertwined with association areas (Buckner et al, 2011). But since it is out-of-scope here, I'll leave it at that much.
See it like this:

For a more detailed mechanism and circuitry of the cerebellum, you can have a look at this interactive page from the University of Texas. @FilipeRocha have also come up with some fine details in their answer.

References:

Ghez C, Fahn S (1985). &quot;The cerebellum&quot;. In Kandel ER, Schwartz JH. Principles of Neural Science, 2nd edition. New York: Elsevier. pp. 502–522.

Buckner RL, Krienen FM, Castellanos A, Diaz JC, Yeo BT (2011). &quot;The organization of the human cerebellum estimated by intrinsic functional connectivity&quot;. J. Neurophysiol. 106 (5): 2322–2345. doi:10.1152/jn.00339.2011.