Laser and galvo project
What kind of wires (input jacks) do I need to buy for laser and galvo circuit boards (designated in the images above)?


A quick Google search turns up a few guides to wiring up that board, or at least one that looks just like it.  I suggest taking a look at one of those.  But in general, assuming that is actually a polygonal scanner and not a galvo, it is going to take a DC voltage (probably 12, 15 or 24v), and a clock signal, which the driver will use to synchronize the motor rotation rate to.  The clock signal is used to let you synchronize the beam scanning to another axis (for 2D scanning) or to a power modulator (if you are drawing something with the beam).  There is probably a feedback signal as well that indicates the true phase of the mirror so you can figure out if it is locked to the clock and/or detect error conditions.  

Since you don't know how many clock edges corresponds to one revolution, I would start by trying to get the motor spinning at a relatively low speed and then use a laser and a photodiode to measure how many times per second the beam scans.  Since the mirror scans 720 degrees per revolution and looks to have 4 faces, from that you can calculate how many clock cycles there are per revolution.