I’m creating a boot drive for my laptop to run Ubuntu Mate off of from BIOS. I got the ISO file from the Ubuntu MATE website, what else has to be on the drive for BIOS to recognize it as an operating system and run it?

Hi Nothing else needs to be on the drive. This is why you can burn it to a DVD and have the DVD boot. In the same fashion, you can burn it to a USB drive using the very low level dd command.