Sorry to here about your problem. A few questions.
Does the computer beep at all? As this might give us a clue about whats going on.
Are you happy to open up the computer? If so it may be worth reseting the bios. This is usually done simply by moving a jumper. (Sorry, I cant be more specific unless I know you computer model.)
Did you install any new hardware that could be causing the problem?
Try unplugging any usb devices that you have attached as this can sometimes mean a failing power supply.
Super cool's good advice about unplugging leads and replugging them in sometimes works and its definitely one of the cheapest things you can try, lol.
Does the computer display anything before th HP screen?
I think thats it for now, unfortunately it could be a number of things, hard-drive, power supply, processor, memory or even the motherboard. Apologies if you have tried all this already.
Rich