Root Samsung Galaxy SIII

There are quite a few possibilities as to why Odin would fail. For one thing, are you sure you're flashing the file specific to your S III model? Each carrier in the US has a different protocol, and the international model is a whole different animal. Make sure you look up your device's model number in Settings > About Phone first. Also, be sure you've downloaded the latest Samsung drivers for USB connection.

Assuming you have the international model, you can go to XDA Developers and follow the directions in the linked video to achieve root.

If that doesn't work, you can try Casual but make sureyou're using the right version of the program, because you can very easily destroy (brick) your phone if you have the wrong firmware or carrier/model number. A quick Google search should help you find the right version of Casual. I think they make it for GSM and CDMA carriers.