What version of the S3 (e.g., what carrier, or unbranded international) do you have. What update did you try to install, and what instructions did you follow (post links). It's never a good thing to manually apply an update if you don't know for sure what it is. If you're lucky, the ROM didn't actually install, and a factory reset will fix it:
Turn the phone completely off (this won't work if you start with the phone on). Press and hold volume UP (not down)+home (not menu)+power. Hold all three until you see the Samsung logo, then let all three go immediately. If you get it timed right, the phone will boot into recovery. If you release the buttons a little too soon or too late, it will try to boot normally. Just turn the phone off and try again.
Once you're into recovery, use the volume buttons to select Factory Reset, and the home button (or maybe it's the power button, I don't remember--if home doesn't do anything, press power) to execute it.
If you can't get into recovery at all, or a factory reset won't help, you're going to have to try to boot into download mode, and try to load a complete, stock factory ROM. To help you we'll need to know what version of the phone you have.