My screen of my sony vaio computer is black

You would get more help if you start your own thread.

I will help you anyway, but I am just letting you know that you might get more and better help from others if you make your own.

Anyway, first things first, is that you want to get that data from you laptop backed up and safe since it seems important to you.

You said the lights come on and the fan makes noise, so there's a chance that the Hard Drive is fine, but you just aren't getting any display from your LAPTOP's screen. So what I would do is connect your laptop to an external display (like another computer monitor or TV screen). Many laptops have a HDMI or VGA port so you can use an external display (you will need a VGA or HDMI cable). If you get a signal on a different display, we can rule out that there is something wrong with your graphics card and that it's just your laptop's screen that has died. So if you can get picture on another display, you should be in good shape and be able to back up all your data onto an external hard drive or USB flash drive (depending on much space you need).

Secondly, if you can't get display with an external monitor, you can should buy an external enclosure (most laptops use 2.5") and remove your hard drive from you laptop . Then, you would use the enclosure with your laptop's HDD inside and back up all your data on another computer (like a family member's computer or somebody else your trust). Or you could just use that computer to copy the stuff you want to keep from that Hard Drive to another storage device.

As far as repairing the laptop's screen or the graphics card (if that is the problem), I really wouldn't know how to help you here. I had a driver issue with my laptop before where I updated the driver and my screen just gave me black. I had to use a restore point to go back (I had to connect my laptop to an external display). So you can try restoring your computer or doing a factory restore once you got all your data backed up.

If it's an old laptop, it may not be worth getting it repaired (a shop will likely tell you the screen can't be repaired but has to be replaced which can cost 100-200 dollars). So it may not be worth spending that much if you have a laptop more than 4-5 years old since you can probably buy a better laptop for 400-500 dollar with better technology.