A fingerboard in arcade terms is a small circuit board used to convert a PCB from one pinout to another. Fingerboards are usually designed to fit the standard JAMMA harness, although other pinouts are also available.
Lets say for example you have a Dig Dug arcade game board, and you would like to install it in your JAMMA compatible arcade cabinet. You would buy a JAMMA fingerboard, and then you would solder wires from each of pins on the Dig Dug edge connector to the equivalent pins on the fingerboard. You can then plug the fingerboard into your JAMMA harness, and play Dig Dug to your hearts content.
Higher quality fingerboards require less soldering (or none if you also have the wiring harness for the game you are converting), as they will have screw tag strips on them. These fingerboards can also easily be reused, since you don't have to permanently attach anything to them.
Most titles can be easily converted to the JAMMA standard with a fingerboard, although some titles require custom fingerboards that do power conversion (or video inversion in the case of Donkey Kong).