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Well, I dont quite agree 100% with vlsi_freak. Because every software runs on hardware, weather its software firmware or embedded s/w.
Not every hardware needs a controller. And not every controller needs a firmware/sotfware.
Firmware is the software which is not as soft as software. Well, it is actually embedded sotfware. Well it is actually that embedded software which usually resides in ROM. So its firm, not soft. i.e it cannot be changed once put in ROM.
So its kindof something between software and hardware. Not as hard(fixed i.e which cannot be changed once manufactured) as hardware, and not as soft(flexible, i.e can be changed) as sotfware.