The first thing I would try would be run System Restore. If that doesn't fix the problems it is probably because of damaged hardware--your video card or motherboard. If you have onboard video then it may be a bad motherboard. It's common for capacitors to rupture when power goes out like that--and a few years ago there was a large batch of caps that were bad.
You might open your case and check the capacitors. If you don't know what they look like, there are some photos in this Wikipedia article--altho they show side view when most will be standing on end.http://en.wikipedia.org/wiki/Electrolytic_capacitor
If you see any caps that are obviously busted or have a little mound on top (instead of being flat and shiny) then you've got bad caps and will probably have to replace the motherboard. If you look real hard you might find a place that will replace just the caps, but most computer repair shops won't bother.
I hope I'm wrong about this, but if it is damaged hardware, you should consider investing in a UPS (Uninterrupted Power Supply) system so it doesn't happen again.