Most likely HD44780 compatible. Based on the PCB, my guess would be (what I call pin 1 is the one on the left in the second picture you provided with the component side down):
pin 1: ground
pin 2: contrast adjust
pin 3: Vcc
pin 4: Vcc
pin 5: ground
pin 6: RS
pin 7: R/W
pin 8: E
pin 9-16: D0-D7
One of the grounds and one of the Vcc pins will be for the backlight (if available). Putting 5V across them with say a 100 ohm series resistor should make the backlight light up. No guarantees for correct polarity, though
. If pin 5 is connected to pin 1 (the ground plane), then I would say my guess for the power pins should be correct. No way to be sure about the data lines (I assumed standard pinout) without trying though. There are plenty of Arduino sketches to drive an HD44780-compatible LCD, but I would suggest to start with a display with a proper datasheet if this is your first time driving an LCD. That removes a bunch of unknowns from the equation. For example, it's possible that this LCD is not even HD44780 compatible.
It would be useful to have the board in the printer it was connected to.