This is How you root your Samsung Galaxy S3, please check your model number before downloading the respective Recovery files.
A custom recovery allows you to flash custom Roms, root your device and install the latest available android operating system on your device.
Read further for more info.
Check Your Model number
AT&T (Use this for Canadian SGH-i747M) –Download
US Cellular –Download
S3 Mini –Download
Please Follow the Instructions carefully!