My Computer Tower’s lights are coming on, but not my Computer Monitor: What is the problem?
There are usually only two things that can be at fault with this error: the Computer Monitor itself; or the Graphics Card on your Computer (the piece of hardware that controls the visuals on your Computer Monitor).
The easiest thing to test is the Computer Monitor.
Basically, take the Computer Monitor, unplug the Computer Monitor’s cable from your Computer Tower, and then plug that same cable into either someone else’s Computer Tower, another Computer Tower you have in your house, or into the Graphics Port on a Laptop (which looks something like this – 
). Turn on the Computer (or the Laptop), and if the Computer Monitor comes on when it is plugged into a different Computer/Laptop, then it’s not your Computer Monitor that’s at fault – it will be the Graphics Card on your Computer itself. If the Computer Monitor doesn’t come on when it’s plugged into another device, then I’m afraid it’s your Computer Monitor that is broken, and you will need to get a replacement.
Another thing, regarding the Computer Monitor, that it could be, is the Computer Monitor’s cable that connects it to a Computer Tower or Laptop. Obviously, the simplest way to check the cable is to either attach it to another Computer Monitor and Computer Tower/Laptop, or to try a different cable with the Computer Monitor that isn’t working. If the cable works on another Computer Monitor, then there’s nothing wrong with the cable, and therefore if a different cable works with the supposed broken Computer Monitor, then it’s the original cable that is broken. As a result, depending on whether the Computer Monitor is actually working when it is attached to a difference device, then we now need to check the Graphics Card.
Many Computer’s nowadays come with Graphic Card’s that are physically built onto the Motherboard of the Computer. These are usually referred to as On-Board Graphics or Integrated Graphics. There are two ways to check whether you have On-Board Graphics or not – one is to check your Computer’s documentation, be it either in the Computer’s Manual, or by searching for the make and model of your Computer on the Computer Manufacturer’s website (obviously on a working Computer or Laptop). For example, if I search on Dell’s Website for the Dell Inspiron 570, it brings up this webpage, and on this webpage it tells me that my Dell Inspiron 570 has Integrated ATI Radeon HD4200 Graphics.
The second (and probably easiest) way to check whether you have On-Board or Integrated Graphics is to turn your Computer Tower round and look at the back. If you see anything that looks like the following picture: 
– Then you have an On-Board/Integrated Graphics card.
If you don’t see anything like the above description and picture, look a bit further down the back of your Computer Tower, and if you see something that looks like the following picture: 
– Then you have what’s known as an PCI-E (Peripheral Component Interconnect Express) or AGP (Advanced Graphics Port) graphics card.
If you’ve deduced that it’s neither your monitor nor your monitor’s cables that are at fault, then if you have a PCI-E or AGP Graphics card, then the easiest thing to do is to try a different AGP/PCI-E card in your AGP/PCI-E slot on your motherboard.
This may be the time to get someone to help you with this, or if you’re fine with going inside your Computer Case, then read on 
(Also, if your PC is still in warranty, it is advisable that you phone the supplier/manufacturer of your PC, because going into your PC will void any warranty you have. So please don’t read any further, and phone the supplier/manufacturer instead if you want to keep your warranty.)
Firstly, we need to find out whether you have an AGP or PCI-E Graphics card. If you don’t have the specifications to your Desktop Computer handy, nor can you find the manual on the Internet, then we need to go into your Desktop Computer (we’ve gone away from Laptops, for 99% of Laptops have integrated graphics, so see above for that) and have a look around.
So unplug everything from the back and front of your PC, and take the PC into a medium-sized space, usually a living room or on a bed on a bedroom. Place the PC case/tower on its (staring at it from the front i.e., where your DVD/CD drive is visible) right side, and take off its left side.
(In several, although rare, cases, the side that comes off is the right side, and not the left. Although these cases are few and far between, you should make sure that you’re taking off the correct side, which is usually the side that has air holes in it. If the side you’re taking off doesn’t have any air holes in it, then you’re taking off the wrong side.)
Don’t be afraid to do this, for the side of a PC case is just a slab of metal, and isn’t attached to anything important on the inside of your PC. Most cases have the side held on with screws which easily come off, although several manufacturers (Dell and HP noticeably) it is a screw + latch which you will need to either push down or up. Some may even only have two latches that you need to move away from each other at the same time in order to get the side off.
Once the side is off, you should see an assortment of different cables attached to the inner components of your Desktop Computer. Before you touch anything, grab hold of the sides of your case/tower with both hands. This is to ground you, and remove any static that you may have. It is also usually advisable that you use one hand to hold onto the case/tower at all times to keep you grounded and prevent the build up of static, although this isn’t technically necessary.
Looking at the inside of your PC, it should look something like this: 
– The metal component with the fan on it is your heat sink, which covers your CPU and keeps it cool so it doesn’t over heat, and below your heatsink will be your Graphics Card.
Look further in, and check how it connects to your motherboard (the largest component on the inside of your Desktop Computer that all the other components connect to). If it looks like the following: