Camera firmware is a DSLR's basic software and coding, which the camera maker installs at the time of manufacturing. The software is stored in the "Read Only Memory" (ROM) of the camera, and so it is unaffected by battery power.
Firmware is responsible for actually making your camera work, and it is therefore vitally important. From time to time, camera manufacturers will release firmware updates, which will upgrade the camera by enhancing performance, or by adding new features. It's important to check for firmware updates from time to time.
Firmware updates are installed by using a computer to download any updates onto the camera from manufacturers' Web sites.
Although firmware updates are designed to improve the functionality of DSLRs or any other type of digital camera, they are not obligatory, and some minor updates may be completely pointless, such as, for example, adding a language to the menu system that you don't even speak!
Care also needs to be taken to ensure that the update will actually work on your existing camera. Some updates require a certain level of firmware to already be installed, and others are "region" specific, so you need to make sure you are installing the firmware for the North American region, and not a region elsewhere in the world by mistake!
You should also bear in mind the way that your camera uploads new firmware. Some cameras have Programmable ROM (PROM), which allows new information to be added to the system. Others have Electronically Erasable PROM (EEPROM) which allows information to also be erased. These cameras are obviously preferable, as you are not stuck with firmware updates if you don't like them.