FIrmware is the layer of software below the operating system. In computer motherboards, it is called BIOS or UEFI. For phones, printers, graphics cards, network cards, and everything else, it is simply firmware. It's what actually controls your device, and tells the components how to interact with the software.

Why does this matter? Like everything else, firmware can have security holes. FIrmware updates patch those. In addition, they can give more functionality, and improve performance.