Hi, 'booting' is just a word which means powering on a computer in a nutshell.
When you 'boot' a computer, it starts BIOS(Basic Input/Output System) which checks all hardware components(hard drives, floppy drives, etc.) and ensures that they all work. It also has a lot of other functions, for more information, click here.
Device drivers are files which are needed by the operating system in order to recognise the hardware present and required for it's correct functioning.
For more information on drivers, click here