Why is my laptop's display not working?

Have you tried connecting your laptop to an external monitor to see if it will display? If it does, you either have a bad screen or ribbon cable connecting from the LCD to the motherboard.

If it doesn't display on an external monitor, there's an issue with the graphics chip on the motherboard. What happens overtime is heat/various other unknown factors cause the solder connection with the board to weaken. In this case, your laptop's graphics chip will need to be "reflowed." This process requires a heat gun to heat the chip to a temperature where the solder underneath it melts and regains contact with the board. However, success from this procedure isn't always guaranteed and even when it's successful, there's no telling how long this will last as it is only a temporary solution as the chip itself isn't healthy. I've had cases where this solution extended my laptops life for a couple years and cases where it only extended it by a couple of days/weeks.

Take it to a local professional and tell them that it needs a reflow. Many technicians know what it is.