Firmware
Data stored in a computer or other hardware device read-only memory (ROM) or elsewhere in the circuitry that provides instruction on how that device should operate. Unlike normal software, firmware cannot be changed or deleted by an end-user without the aid of special programs and remains on that device regardless if it's on or off.
A firmware update is a software program that can be used to flash update the firmware in those devices. For example, a user could download a firmware update for their disc drive or network router to update it's capabilities or fix a issue with that device. Firmware updates are made available through the hardware manufacturer and in some cases are available through computer enthusiast websites to give a device even more capabilities then the manufacturer intended, although these updates are not supported by the manufacturer.
Also see: BIOS, Bricked, EEPROM, EPROM