There is no visible graphics output

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Minor: Best case, you twiddle the cables or open the PC to reseat the card, and fix what had been a poor connection. If you just hooked up the monitor, make sure that you plugged it into the correct port. Many graphics cards sport both analog VGA and Digital Video Interface output connectors.
 
Moderate: If the simple corrective actions fail, check for heat buildup from a failed graphics chip fan--especially if the screen goes dark after a period of operation. Again, the Motherboard Monitor utility provides a real-time display of case temperatures. Also, after allowing things to cool a bit, open the case and boot the PC; then check whether all fans are working and all vents are clear.
 
Dire: If these steps fail you, narrow down the source of the problem by plugging a different working monitor into the card. If the new display is blank as well, you may have a fried graphics card. Another clue: If at boot time you can't see even the POST output (the text that displays before Windows loads), the problem could be serious. Of course, if the second monitor works perfectly, the trouble is almost certainly in the display.