After having switched on my PC the monitor did not come on. I thought I must have switched it off by mistake. So I pushed the switch and it still did not come on. On switching once more the green light appeared for a few seconds only to go back to the 'off' position. Is this a sign that the monitor has packed up?
Then as onthelimit1 says it is either the graphics card, the cable connecting the card to the monitor or the monitor itself. You say that the light goes green for a short time and then goes to the off position. By that do you mean the light goes off completely or to its standby colour e.g. orange or red?
Then you have to start with the obvious and check the cable connections between the graphics card and the monitor - give them a good wiggle and watch the screen, tighten the connection on the ends of the cable if necessary. Then if that fails look at the back of the PC to see if your graphics comes from the motherboard or a separate graphics card. If the former the socket with the video lead will be along the edge where your power, USB, mouse/keyboard come from. If you have a separate card it will be several inches higher and come from one of the PCI slots.
For graphics on the motherboard there is nothing that you can do but borrow a monitor to confirm that it is dead. For a separate graphics card you can take the case off the PC, remove the graphics card, reseat it and try again, you could also check if the graphics card fan is spinning (assuming it has one). But here again having a known good monitor to test it with would be a distinct advantage.