How to root / Jailbreak Samsung Galaxy S3

Install CWM ClockWorkMode

Step 1: Download Odin 3.04 On your computer. Download Samsung USB Kies Driver Download clockworkmod.tar.

Step 2:  Put your Galaxy S3 into USB debugging mode by going to Settings => Applications => Development => USB debugging. Now turn Off your phone and just remove and reinsert the battery.
Step 3: Put your Samsung Galaxy phone into download mode: Press and hold  Volume Down + Home + Power keys simultaneously.
Step 4: On your computer Run the ODIN application which you downloaded in step 1.
Step 5: Connect your device to your computer using the USB cable. If you have completed the above steps the Odin window will show that the device has been detected otherwise it will not respond.
Step 6: In the middle of the ODIN screen click the PDA button and select clockworkmod.tar and press Start Button.
Step 7: ODIN will start the process of flashing your Galaxy S3 firmware. The phone will reboot and the flash process is complete.
Root / Jailbreak Samsung Galaxy S3

 Step1:Download the software CMW_SuperUser_v3.07.zip

Step 2: Connect your device to your computer or directly copy CMW_SuperUser_v3.07.zip file into Galaxy S3 external SD card.
Step 3: Turn Off the Galaxy S3.
Step 4: Put your device in recovery mode. Press Volume Up + Home Button + Power Button until your Phones boot into CWM Recovery Mode.
Step 5: Once your device is in recovery mode keep pressing the volume down button until you find the option “Install Zip From Sd Card”. Press Power button to select the “Install Zip From Sd Card”  option.
Step 6: Select the copied file after you select the  option “ Choose zip from SDCard”.  Scroll down by pressing volume down and then Power button to choose “CMW_SuperUser_v3.07.zip”
Step 7: Select the option “Yes- Install CMW_SuperUser_v3.07.zip”. The installation will be completed in few moments and you will need to reboot your Galaxy S 3. Now it is rooted.
If you want  to check your Samsung Galaxy S 3 device is properly rooted or not you can download  root checker tool from google play store...

Tutorial