Hi, I'm a complete novice to this, but I do have some limited knowledge of electronics. I need to run 2 sets of ws2812 strips from 1 5v 40a power supply. I will be powering the Arduino board separately. What I haven't been able to find is how to power and control these strips.
I've attached a crude diagram of how I think it might be done but could really do with some advice.
Thanks
OK, well, firstly if you have a regulated 5 V power supply, it is a bit silly to use a 12 V power supply to power the Arduino - and it is generally unwise to power it via "Vin" or the "barrel jack" in any case. 
You reference to "data pins" and the different strip lengths indicates that you are using two different pins to drive the two strips which is appropriate. A Nano can drive up to 300 individual WS2812 LEDs however distributed. UNOs are impractical for most serious projects such as this.
Now as to the arrangement of power and data cabling. Clearly the power - and that always includes the ground connection - must always go directly from power supply to LED strip. And it is important that not only do the power and ground run as a pair, but also that the data feed travels - again as a pair - with the ground, wherever it goes. Duplication of the ground wires is not a good idea.
So what you need to do is to cable the two power wires and the data line from each strip, back to the power supply terminals where the data lines then pass to the Arduino along with 5 V and ground - the 5 V going to the "5V" pin of course. It would be a very good idea to put a 1 A fuse in the 5 V line to the Arduino for two reasons. One is in case you accidentally shorted something else connected to the Arduino; yo do not want it going up in smoke.
The other is that you can remove the fuse from its holder whenever you need to connect the Arduino to a computer via USB - such as for programming.
Do not forget the 1 mF capacitor across power and ground at the start of each strip, the 330 Ohm resistor in series with the data line at the same point, and to run the power cable also alongside each strip to power it at both ends and "tap" it in at least each 100 LEDs.
Thank you for your response. As stated I am a total novice to this wonderful world of arduino and to be honest it is proving difficult to grasp as this is my first project ( trust me to start with a hard one