DETAILS
If the screen powers up (screen light goes on):
If the screen draws it's supply from the PC desktop or tower but remains black, then in order of probability it is likely that::
Note: If the screen powers up, does not report memory or reports memory and then immediately hangs up, it is likely you have CPU (processor) problem.
If the screen powers up with memory reported and it goes on to report other details then waits - leave the PC alone until you get a message. It is likely that the BIOS has lost hard drive settings or the hard drive has failed (primary hard drive failure), or that the floppy drive has failed, due to bad connection of it's data cable.
If the fan is blowing hot immediately after power up, replace the Power Supply Unit.
Note: On an ATX case the main power switch (hard power) is on the Power Supply Unit with the front power switch on the PC case being 'soft power'. There is a soft power jumper on the motherboard which if not shorted by a jumper, will disable both the soft and hard power switches.
If the screen light goes on, the power fan is blowing, the memory is OK and the floppy power cable is on 4 pins, then it is likely to be a motherboard fault, or less likely a power output fault from the Power Supply Unit. If you are able, use a separate Power Supply Unit you know is working, attach it to the motherboard and if have no results, it's the motherboard.
If the screen does not power up (no screen light):
If it has an independent power cable check firstly check to see that it is fully in at the back of the screen, then check the cable's fuse and/or remove the cable and check it on another device e.g Electric Kettle. If the cable is OK, then if you hear the hard drive of the system unit (The PC Box) being accessed /see the drive light flickering, then it is likely that the screen is faulty. If the fan is not blowing, it's the Power Supply unit..
If the screen power runs from the PC Supply Unit, then firstly check that the cable is correctly in at both the PC and screen. If the cable is fitted correctly and the PC Supply fan is not running then the Power Supply fan or the whole Power Supply is faulty, if the power supply is running then it could be the screen that is faulty. Before assuming this as fact, try the screen on another PC.
Note: Never run a PC for any substantial period when you know it's Power Supply Unit fan is not working.
Next: a cautionary note on PCs originating in other countries:
Power Supply Settings - Voltage in Different countries
United Kingdom:
In the UK we use a 230/240v supply and 3 pin plugs, ensure that:
United States & most other counties
Countries other than the UK usually have 115v supply with a 2 pin plug. Again a normal case should only have a 5amp fuse in the power lead plug. If you have any doubts, consult your local PC shop or electrician.
ALWAYS CHECK THE LOCAL POWER SUPPLY RATINGS BEFORE STARTING UP A PC WHICH WAS MADE/SETUP IN ANOTHER COUNTRY. For example a PC from the United States: At the back of the case you should find a power rating switch, ensure that this is flipped up to 230v for UK usage. Failure to do so WILL result in hardware failure and the cost to you of replacing components.