Start eliminating things, what’s left when you are done is what’s likely causing the problem.
Here’s the starting list:
– Problem with the monitor or its components (power and data cable)
– Computer or its components (video card or motherboard)
The green light is on, that means the monitor is receiving power, which means it’s not the power cord between your monitor and the wall. Replacing the data cable would confirm or eliminate it being the problem.
The easiest way you can take the next troubleshooting step is to plug in a known working monitor to your computer. If that monitor works, then you know it’s a problem with the old monitor, proceed with repair or replacement as @IchtheosaurusRex advises.
If when you plug in the working monitor, you still have the problem, then it is likely the graphics card (or another component inside the computer) that is the problem. If you are not computer hardware-savvy, it becomes pretty hard to keep troubleshooting from this point on because it requires getting inside and replacing computer components, so I would advise bringing it into a mom-and-pop computer store (don’t go to Best Buy, their “geek” program is seriously flawed and not worth the prices they charge).
If you can’t get your hands on another monitor, I would go straight to bringing the computer (both the tower and the broken monitor) to a computer repair store. They have the diagnostic tools to tell you quickly what is the problem, and they can offer you a quote to repair it. Once you have the problem diagnosed (which may or may not cost money depending on how nice the company is), you can go quote-comparing if you want to.