Sainsmart Omron 12V optocoupled 2 channel relay problem
Lets troubleshoot this circuit:


The circuit diagram you put in your question shows a wire creating a short across the relay coil. Check this, as this will definitely stop your fan from spinning. However, I assume this is an error in the diagram.
The picture on the website shows a jumper link between VCC and JD-VCC. Make sure this is removed.
Check that the fan works when you power it directly from the 6V battery. The fan may work only at a higher voltage or run off of AC voltage. Also, the battery could be flat - measure its voltage when it is under a small load.
Drive the input circuitry directly off of 5V. That is connect VCC to 5V and then directly connect IN1 to GND. Open circuit the IN1 input to turn it off. Check that this direct driving opens and closes the relay contacts.
Check the 12V supply. Does the voltage drop significantly when the relay pulls in or out? Does it have enough current? "Each [relay] needs 15-20mA Driver Current" Measure it using your DMM in ammeter mode.
Check your wiring is correct. It sounds simple, but you will be surprised by how many times the wiring is the cause of the problem.
Measure the resistance of the relay contacts in both the open and closed states. Arcing or poor manufacturing can be causes for the contacts to not function as intended.