How do I roll back from Yosemite to Mountain Lion without Internet Recovery after erasing my hard disk? I erased my MacBook Air hard drive - which came with Yosemite OS X - because I wanted to roll back to Mountain Lion (OS X 10.8), since Yosemite was giving me problems with my files from the previous MacBook Pro.

I deleted the drive using Disk Utility. While partitioning, I forgot to make a GUID partition table. I tried installing Mac OS X on an USB-stick. It's not showing my drive, but just my USB Mac OS X installer. When I click on it, it shows the stop sign.

How can install Mac OS X without using the Internet Recovery for Yosemite
If you're trying to boot from a USB OS X installer and it shows up as a bootable device but then proceeds to show the stop sign and halt, this indicates the installer will not run on your Mac as your system is unsupported by the startup files on the installer.

As klanomath points out, if it shipped with Yosemite it's unlikely to support Mountain Lion. Your Mac may support Mavericks (which is more akin to ML than Yosemite) depending on the model family - and obviously also depending on whether you can get hold of the installer for 10.9. If you can get hold of a fresh copy of the 10.9 installer from Apple this will be the last iteration, 10.9.5. Importantly, this last version will boot all Mavericks-capable Macs up to and including all 2014 models. If this includes yours then Mavericks should install as long as it's 10.9.4 or later.

Your existing drive won't show at this stage as it doesn't have an OS X installation to boot from, even if it's correctly partitioned.