First put it on the charger for 24 hours - it could be an almost discharged battery. If that doesn't fix it, bring it into a shop and ask them to try the battery in another phone - it could still be a bad battery. (Don't panic - you can buy a new battery on Amazon for WELL under $10.)
If it's not the battery, it could be a bad app or SD card causing it. If you have an SD card in it, take it out, then try again. If it's a bad app, boot into safe mode (see
How to boot the Galaxy S3 into Safe mode? ).
At very worst, if it's not a hardware problem (that requires a shop, parts, labor and it might not be worth it worth an S3) it may beed the ROM reflashed. Again, it's something you can do - see
[Samsung] How to flash Stock ROM via ODIN. (Try not to die of boredom after the flash completes - it's 10 minutes of watching grass grow in the winter.) Back the phone up first (
Backing up an Android Device) because you'll lose every change you made since you got the phone - texts, apps you installed, data - everything.