Kernel is one of the most important part of Operating System. We use the word kernel to mean the part of operating system that runs in the privileged mode (Sometimes even a subset of this). Kernel is closer to the hardware and often performs tasks like memory management and system calls.

Now for BIOS (Basic Input-Output System), it is the one which is responsible to provide drivers for new devices to OS. BIOS constitutes of the code that is stored in read-only memory (ROM) and some configuration data in non-volatile RAM.
BIOS provides three primary functions:

1. Power on self test (POST), so it knows where to load the boot program.
2. Load and transfer control to boot program .
3. Provide drivers for all devices.

The main BIOS is supplied as a chip on the motherboard. It contains everything needed to perform the above three functions. Additional BIOSes on other boards can provide access to additional devices.