Why is my relay not triggering?
From the image in the question, the jumper across pins J5 (in the
below schematic) or JD as on the silk screen on the PBC is missing.

Using the above schematic, a choice to power the Ardino and the relays off of one 5V power supply can be made.  This appears to be done by connecting the 5V to the VCC pin, Ground to the GND pin &amp; placing a jumper across J5.  However, it appears that the relays can be isolated from the Arduino if desired.  This appears to be done by connecting the Arduino's 5V to pin 2 of J5.  That should be enough to light up the LED indicator and therefor the optical isolator when the Arduino's output to IN1, 2, 3 or 4 is low.  Then, connect the 2nd power source 5V to pin 1 of J5 and the ground from the 2nd power source to pin GND.  This last step should provide the power to close the relays when the Arduino takes IN1, 2, 3 or 4 low.
Lastly, make sure the polarity of the 2nd power supply (if one is used) is correct.  The output side of the optical coupler and the transistor driving the relay only work when the current is passing in the proper direction. Not to mention, the reverse bias flyback diode (D1, 2, 3 &amp; 4) would short out the power supply if connected in the reverse direction.  However, this catastrophe is avoided here as the transistors (Q1, 2, 3 &amp; 4) would not pass current in the reversed direction.