What is the difference between BIOS and firmware?

Could anyone elaborate on the differences between BIOS and firmware please?

Solution:

As others already stated, BIOS is the specific name for the (motherboard) firmware in older PCs. New computers these days have a technically somewhat different kind of firmware which is called either EFI or UEFI.  

Please note that any computer will contain, besides the BIOS (or EFI or UEFI), also other firmware. Network cards, video cards, RAID controllers, hard drives, flash drives, SSDs, sound cards, just to name a few examples, can all have firmware embedded inside the device.

Weirdly enough, the firmware of a video card is often called the video BIOS. This is technically incorrect. BIOS is appropriate only for the startup firmware of the motherboard itself.