Yes, microcode is basically firmware that runs on the processor. The special term "microcode" specifically refers to the firmware on a processor that contains the blueprint for translating from standard machine language to low level processor instructions. So it is a more specific term than firmware.

Microcode simplified the job by allowing much of the processor's behaviour and programming model to be defined via microprogram routines rather than by dedicated circuitry. Even late in the design process, microcode could easily be changed, whereas hard-wired CPU designs were very cumbersome to change.