These are the instructions to root Samsung Galaxy Nexus I9250 (running on Android 4.1.1 and 4.3) using Odin on Windows Computer.
Remember: Your Samsung Galaxy Nexus should have at-least 40-50 percent of battery to perform the rooting process.
Step 1: Download and install Galaxy Nexus USB driver on your computer. If Samsung Galaxy Nexus USB drivers are already installed on your computer then skip this step.
Step 2: Enable USB Debugging on your Galaxy Nexus Smartphone. To enable USB Debugging: Open Settings > Developers Option > USB Debugging > Tick to enable.
Step 3: Now, Power off your Samsung Galaxy Nexus Smartphone.
Step 4: Now, you have to enable Fastboot Mode on your smartphone while it is power-off. To enable FastBoot Mode: Press and Hold Volume Up, Volume Down and Power Button at same time until FastBoot Mode is enabled.
Step 5: Download and extract Root files on your Windows Computer. After extracting you will be able to see the following files:
Step 6: From the extracted file, Open root-windows.bat.
Step 7: Once root-windows.bat is opened you will see the following window:
Step 8: Press any key from the keyboard of your computer to begin the rooting process.
Step 9: Now you will be able to see Unlock Bootloader Screen on your Smartphone. Select Yes using volume up/down button and power button to continue.
Step 10: Now, on the next screen you will be able to see the FastBoot Screen with Disabled Lock State (at button) which means you have successfully unlocked the bootloader of Galaxy Nexus.
Step 11: Now, Galaxy Nexus will automatically show red android logo and rooting process will continue. Once Rooting process is completed your Smartphone will restart automatically.
Step 12: Your Google Galaxy Nexus smartphone will take 2-3 minutes to restart. Once your smartphone is restarted successfully, you will be able to see the welcome screen.
Step 13: Now your Galaxy Nexus is rooted successfully. To confirm, Open App Drawer (menu) and you will be able to see a new Application installed called SuperSU. If you find this app then it means your device is rooted.
Optional: You an also verify the Root Status of your Samsung Galaxy Nexus Smartphone using Root Checker Application.
[*] No data loss process: Your Data like contacts, pictures, videos, music and etc will be not affected with this method.
[*] Warranty Void: Your Samsung Galaxy Nexus Smartphone Warranty will be Void after following the above tutorial.
[*] Google Nexus Pixel Stock Firmware: If you are looking for the original firmware for your device then head over to Download Google Nexus Pixel Stock Firmware for all devices page.
[*] Google Nexus Pixel USB Driver: If you are looking for the original USB Driver for your device then head over to Download Google Nexus Pixel USB Driver for all devices page.
[*] Credits: The Samsung Galaxy Nexus Root Tool was created by Chainfire, so full credits goes to him for the root tool.