Here's the most common issue I run into with these little laptops when people bring them in: The power supply.
1. Flip the computer itself upside down and look for what voltage and amperage it's rated at. I think it's 10.5v by 1.9A. Now look at your charger, does it have those same specs? If you have an after market charger, it may have been lumped in with it simply because it fit in the jack, which stinks.
If the voltages/amperages do not match, take it straight to a repair shop and see what they can do. Something's been fried. Even under powering it can cause massive issues.
2. Make sure your power supply actually works. If it has a light on it anywhere that used to turn on when you plugged it in, look to see if it still does. If this doesn't help, take the charger to an electronics repair shop and have them test it. If none exists, cross your fingers and go into a Radio Shack. They have all the equipment needed to test this, but they may or may not do it for you - it entirely depends on their mood that day.
3. If it's out of warranty, or whatever, and you think it might be worth putting a few dollars into, pick up a generic charger for it on the cheap just to give it a go. I found one here
Amazon com Super Power Supply AC DC Laptop Adapter Charger Cord for...
for about twenty bucks. I also found a car charger that was 8 dollars cheaper (we're just looking to see if the adapter was the problem after all)
UpBright Car Adapter For Sony VAIO Lifestyle VGP VGN Series VGN P530H...
Look around more yourself and you will most likely find one of these even cheaper than I did, I only spent about 20 seconds looking.
4. If the power supply is not the issue and all the voltages and amperages match up.. Well, unless you've got some cool equipment and mad soldering skills (which you may very well have! I don't know) you could be in some trouble. Check around town for a computer repair shop that works on laptops. Compare prices with what Sony tells you over the phone for their own "in house" repair center.