Firmware is software that makes the hardware do what it is supposed to do that runs on the hardware itself.

The term is used somewhat ambiguously though.

• Sometimes it is just the pure functionality, "if button is pressed, flip the lights on/off".
• Sometimes it glues communication with the functionality, "if signal is received over some interface, flip the lights on/off".
• Sometimes it has an operating system, "when power is on, initiate communications with hardware and interfaces and load software if it is present to interact with any of these".
• Sometimes it is a package with both operating system and software, "when power is on, initiate communications with hardware and interfaces and load software that I know is present".
• Sometimes the OS and/or software in the firmware package has a helpful front facing user interface.