Feeble click in relay but it doesn't get activated when connected to Arduino circuit
You need to join the ground of the external supply to the ground being supplied from the Arduino.  The 12V supply must be capable of at least 100ma

There appears to be a transistor and resistor on the board.  This is probably to drive the relay coil from the Arduino.  However, I would need to look at the circuit diagram of the board to be certain.  Can you supply images of the copper side and component side of the board to see what the connections are.

Ideally the circuit diagram from the manufacturer would be even better.