PC will not start

Does the computer have a DVD device?  Try starting with a bootable CD (Ultimate Boot CD or Knoppix).
If not, see if the manufacturer supplies a bootable diagnostic USB key.

The two-digit hexadecimal code (the "POST code") indicates that the system is failing one of the BIOS startup tests.

We'll need to know the manufacturer of the computer or the motherboard to take it much further than that, as there is no common standard for BIOS failure codes and they vary by model, manufacturer and BIOS supplier.

AMI 2.2 BIOS possibilities are "math coprocessor" or "floppy disk" failure, neither of which makes much sense in a modern computer.

In a Phoenix BIOS, the possibilities would be "initialize hard disk controllers" or "setup power management."

The first thing I'd do would be to pull the HDD, put it in another computer and make a backup of your data.

After that try removing all parts and plugs and reseat them.  Hook everything back up and try again.

If you're not getting anything on the screen at all, and no beeps, then I'm afraid it looks like you're probably going to have to open the case up and start troubleshooting from there.

Is the screen connected to an 'added-in' expansion card in your computer, or the on-board one?

As Dr Klahn has stated, it would be ideal to have further information on the system you've got, to aid in the troubleshooting process.

If you have an external hard drive plugged in via USB, disconnect it and try again. Some of those have been known to disrupt the boot order and make the PC sit like that.

You may also want to open up the case and look for bad capacitors. Could be your motherboard has gone out, as all the lights and fans will come on normally but nothing else happens.

OK, I have been through our Amazon orders and copied all the descriptions of the components making up the system. I dare say this includes unnecessary information:

NZXT Source 530 Full Tower
Corsair CP-9020097-UK VS550 ATX/EPS Vs Series 80 Plus Power Supply Unit
AMD FX 8350 Black Edition "Vishera" CPU (8 Core, AM3+, Clock 4.0 GHz, Turbo 4.2 GHz, 8 MB L3 Cache, 125 W)
ARCTIC Freezer 13 - 200 Watt Multicompatible Low Noise CPU Cooler for AMD and Intel Sockets with pre-applied MX-4 High Performance Thermal Compound
MSI 970 Gaming AMD AM3 GBE LAN ATX Motherboard
2 of Ballistix Tactical 8GB Single DDR3 2133 MT/s (PC3-17000)UDIMM 240-Pin Memory - BLT8G3D21BCT1
(positioned in DIMM 1 and 3)
EVGA 4GB GeForce GTX 960 SuperSC ACX 2.0+ Nvidia Maxwell Gaming Graphics Card
Crucial MX300 750 GB SATA 2.5-Inch Internal Solid State Drive - CT750MX300SSD1

I have tried both sticks of RAM on their own. I have tried different DIMM slots. Nothing helped.

If it will not start at all without the hard drive and no accessory media, then the motherboard, power supply or possibly corrupt BIOS is at fault.

You need to replace these parts or get it serviced. Is it economically worth saving?

Strange question John. Yes, it is. See the list of components.

I have tested (alone) both sticks of RAM from the broken PC in another Dell PC. Both work. I have tried RAM from the Dell PC in the broken PC - it does not turn on.
Nothing wrong with my RAM then.

When I said these parts, I meant motherboard and Power Supply.

Oh, I see. I think we need to know which component is at fault. Then I will just replace that one component.

I have now tested the graphics card in my Dell PC. It worked fine. I put the Dell graphics card in the broken PC and it did not help.

If noting works (no activity) it will be hard to determine between motherboard and power supply.

Can you check the power supply outputs with a voltmeter?

As you are doing nowthenee, it's just a process of elimination, but from what you have described, I'd suspect either the PSU or the motherboard, however I'd trying resetting the BIOS first.

Have you reset the BIOS like I suggested? Power off, then short the jumper pins that are close to the CMOS battery. Also measure the battery's voltage and replace it if it is too low.

There is also a slow mode booting switch on the board. If clearing the BIOS doesn't help try flipping that switch and see what happens.

Yes rindi. While I had the graphics cards out, I took the opportunity to short out the CMOS battery. (The GTX960 was in the way). It was labelled JBAT1. That did the trick. Thank you for that suggestion. So, can you explain to me why the CMOS ever needs resetting? What would cause the CMOS to change?

did you ever look at the article i posted?
it guides you throught the troubleshooting in a short time, to determine what is the cause

You may have been overclocking the system. Sometimes there can be components that can't cope with the speed. If they are running at their limits it is possible that in the beginning everything was OK, but after time those limits are reduced due to "wear" and then things stop working. Resetting the CMOS should change those settings to something like failsafe defaults, and that way it is more likely the system will work properly.

and all components on a pc can be updated - or go wrong
it's not the first time a bios needs a reset - or an update

look on the manufactureers site of the mobo if there are recent updates

I wouldn't risk a BIOS upgrade on that board at the moment, if the flashing goes wrong, your board is bricked. I don't think I saw anything in it's manual mentioning it has a backup ROM from which you could recover from a bad flash. Rather let it run as it is for some time to check it's stability, and then if it runs stable for a couple of days you can try upgrading the BIOS.

I wouldn't risk a BIOS upgrade on that board at the moment  <-- Especially for your own hardware build. I find Lenovo BIOS upgrades to be very safe but that is not this situation.

nobus, I did read that article. It was quite a well laid out article.

We weren't overclocking.

Maybe not knowingly. Those mainboards are built for overclocking and have so many BIOS settings that do that without you realizing it.