Firmware is a program code that is stored in the CPU of a 3D printer's controller. It provides a set of instructions on how the printer works mechanically. Most of the time, the contents of a firmware package uploaded to the controller only need minor adjustments to work properly. These adjustments make minor changes to the values in the source code. These values will specify the hardware and calibrate your particular machine.
Most companies selling 3D printers fully assembled and in kit form provide edited versions of popular firmware for their machines. In most cases, if changes are needed to be made, it is probably better to follow the instructions on their website. Sometimes, these companies don't keep up with the recent firmware upgrades. If this is the case, you may decide to upgrade to a more recent firmware.
Firmware is continually updated and improved upon. There are eleven active firmware listed on the RepRap site at http://reprap.org/wiki/List_of_Firmware.
Here are the two most popular firmware that work with Prusa Mendel designs:
Sprinter (https://github.com/kliment/Sprinter)
Supports printing from an SD card
Supports PID temperature control
Supports acceleration
Marlin (https://github.com/ErikZalm/Marlin)
Supports all of the preceding Sprinter-supported features
Supports look ahead for smoother corners
Supports the Arc feature
Saves to EEPROM
Automatic build platform leveling