Firmware generally runs at the Kernel or HAL. Drivers work in more remote rings (1 & 2) and applications beyond that (3) Obviously, some systems only have two rings, and are pretty monolithic.
In computer terms supervisor mode is a hardware-mediated flag which can be changed by code running in system-level software. System-level tasks or threads will have this flag set while they are running, whereas user-space applications will not. This flag determines whether it would be possible to execute machine code operations such as modifying registers for various descriptor tables, or performing operations such as disabling interrupts