Light Strip Not Lighting

daragh_c:
Can you tell me what the purpose of the relay box is ?

The outputs of the Arduino are 5V and 40 mA maximum current. The relay allows the 5V signal to control the 12V circuit. Either the relay has a 5V coil that requires less then 40 mA or there is a transistor next to the relay which allows the Arduino to switch the higher current or voltage the relay needs. The pictures don't show a clear view of the relay markings or the parts on the relay board so it is hard to tell which is the case.

If you change the Blink example to use Pin 8 (your relay) can you hear the relay click on and off? If not, there is likely something wrong with the relay or the driving of the relay.