A signature error is often caused by trying to install custom firmware (the ROM) with the stock revovery. The stock recovery will only install a factory image for the device, signed by the manufacturer. This means you can only install stock firmware with it. So you need a custom recovery, something like Clockworkmod or TWRP recovery.
Note: do at your own risk, and ensure you have the correct device model for this!
This is the link to TWRP for the Galaxy S3.
This is the link to Clockworkmod (CWM) recovery for Galaxy S3
How to flash a new recovery with Odin:
Step 1: Prepare Odin for Flash
Make sure Auto Reboot and F. Reset are checked, then click the PDA button and load your .tar file.
Step 2: Go into Download Mode on Your GS3
You can do this by powering off your phone, and power back on by holding Volume Down + Home + Power.
If you are on a custom ROM, do not do this by holding the Power button, select Reboot, and select Download Mode. It can cause issues in certain models in Odin.
Step 3: Flash TWRP
Press the Start button on Odin. Then watch your phone do its thing and reboot.
That's it, you now have the latest and greatest recovery available for the Samsung Galaxy S3.
Before installing a new ROM ensure you make a full nandroid backup in your new recovery.
After you install the recovery, boot into it like before, then wipe Cache / Data and install your new ROM.