Firmware traditionally refers to a software component that is not run by a computer in the normal sense, but by a somewhat less powerful computer. In the old days, a microcontroller, the typical single chip or almost single chip computer that received the firmware, was much less powerful that a normal PC. Now things are different and in some applications, e.g. smartphones, the distinction between firmware and software is not so clear, but traditionally the firmware was burnt into the microcontroller memory, in the earliest versions irreversibly, then reversibly as the microcontroller's memory was erasable with ultraviolet light.

Due to the limited resources of the hardware it runs on and to the fact that the distinction between OS and application software is a bit blurred in the firmware arena, normally firmware has a significant connection with hardware and contains interrupt service routines, low level protocols and things like this side by side with higher level components, like GUI or the like.

The adjective "Embedded" refers to the fact that is inside the hardware it controls, not away from it.

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Firmware is the necessary and sufficient amount of software to power up and use the capabilities of a specific hardware comprising of microcontroller(s) and periperal circuits. Firmware typically provides an api to be used by user level software. For example you can have a washing machine design with specific motors, sensors, actuators, displays and keypads. When you have a firmware for this configuration, you can design tens of models of washing machines with different user interface functions and looks.

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Embedded Firmware is a software layer to control and access hardware; provide interface to build application,software on top of that layer. On computer and mobile phones firmware is the operating system.