The most common display problems I see are:
A) No display at all, yet computer looks like its on (power light is on, fan is on, etc)
B) Discoloration or display artifacts
C) Display is very dim
D) Display cuts in and out or flickers when the LCD moves (on a laptop)

So lets go through these one at a time.

A) No display at all, yet computer looks like its on (power light is on, fan is on, etc)
There's two common causes for this sort of problem. The first, is a simple fix. Many times there is simply a static electric buildup in the motherboard or RAM. This simply needs to be discharged. To so so, simply unplug the computer from the wall, remove the battery (if its a laptop), and remove the RAM modules. Then hold down the power button on the computer for approximately 30 seconds. After that, reinsert the RAM modules, reconnect the battery and/or power cable and test it out. If it works, great! If not, then you've got bigger problems.

The second and most common cause is that the video chip overheated. What happens when the video chip overheats is basically the solder holding the chip onto the circuit board gets so hot that it loosens enough for the chip to actually separate from the board, or simply no longer has a sufficient connection to work. If you've ever had a “Red Ring of Death” on your xbox, then this is what happened to it. If this is the case then the most common fix is is a Reflow. Reflow Soldering essentially reheats the chip and solder again along with a “solder paste” and attempts to reseat and recreate the proper connection with the board. Some people like to do “makeshift reflows” using their oven, but unless you really don't care if you destroy the component entirely, just take it to a professional repair shop. This can be an expensive fix however, and it may be worth just buying a replacement video card, or even replacement laptop.

Sometimes that's not the problem either though. If that's the case, then most likely there is a failure with either the motherboard, or the video card/chip itself. At this point I'd say just double check your connections, and test with another video card if possible (laptop users... sorry but you're out of luck there, might be time to get a new computer or drop it off at a computer repair shop and expect a expensive quote).

B) Discoloration or display artifacts
This is usually due to a failing video chipset. Often there's not much to be done but to replace it, if possible. You can try to install the most current drivers for the video card (obtained from the manufacturer's website), and sometimes it just needs to be reflowed as per above.
C) Display is very dim
This is most commonly due to a bad lcd backlight. If you have another monitor to test with, see if that works any better. Most laptops also have a VGA or HDMI video output for you to test with. I usually see this problem with laptops, and almost always with older models. That is because they used to use CCFL backlights (essentially a florescent lamp) to light up the screen. These require a inverter as well, which can sometimes also be the problem. Both of these components can be replaced, but I'd recommend taking it to a professional as the backlight is difficult to do, but the inverter is not so bad. Its also possible to just replace the entire screen. Be aware though that CCFL backlit screens are essentially phased out at this point, and the cost of getting these parts can be surprisingly high. LED backlights are the standard today, and generally don't have this issue.

D) Display cuts in and out or flickers when the LCD moves (on a laptop)
This one usually the LCD cable. There is a cable that connects from the motherboard to the lcd and it runs through the hinge. Its designed to be flexible, but enough opening and closing can sometimes wear it out. Replacing this piece can sometimes be a pain as you need to open up the computer down to the motherboard, and then open up the LCD assembly as well.

I have also seen it where sometimes it an issue with the control board on the LCD itself. Basically sometimes where the cable plugs into the control board gets damaged. Usually the only way to determine (unless its visibly damaged) this is to replace the LCD cable and see if it works.

This sections will be pretty short as there's only a few things to look for.

For desktops, usually when it doesn't power on simply means the power supply failed. This is an easy part to replace, just make sure you get a power supply that is the same wattage as the one you had, and replace it.

For laptops, often its just the AC adapter (charger) has failed. If you have another one, with the same voltage and amperage, you can try that one. You can also try removing the battery, and try turning it on with the battery out of the unit and run off the AC adapter. Sometimes the battery causes a short circuit and just taking it out makes it work again. It can also be an issue with the DC jack on the laptop. This is the port you plug the AC adapter into. Take a small screw driver or something like that and see if you can push the dc jack around. Don't push too hard, just enough to see if it wiggles around. If it does, then its likely that the dc jack needs to be replaced or re-soldered back onto the control board.

For desktops or laptops, another issue that could cause power-on problems is a short circuit. This can happen in many spots, so its usually hard to pin down. A lot of times its the USB ports, where some of the metal leads in the port are bent out of place and touch another bit of metal it shouldn't be touching. You can replace the usb ports, or simply disconnect them if this is the problem. You can also try just unbending the metal pieces, but you still probably won't be able to use the port and I'd rather be safe and disconnect it entirely to prevent it from happening again.

Be aware, shorts can often damage other things as well. You may find that if you find and fix the short, and get the unit to power one, some other things may not work, and may never work. Shorts are bad like that.

Also its worth noting, that the majority of the time I see game consoles not powering on, its either because the power cables are bad, or the USB ports are bad. Rarely do I find it to be something else.