Put simply, the motherboard form factor is the physical size (length & width) of the motherboard, and the standard arrangement of the input and output connections on that type of board.
The motherboard in any computer system is the central component which ties all the other parts together. The motherboard form factor is the single biggest thing which determines how small you can make the physical dimensions of your carputer. It determines the size of case and type of power supply you need to use. It determines how many memory modules and plug-in cards you can add, both of which depend on the number of onboard sockets. The number and type of disks which can be used are again dependant upon the number of onboard sockets.
There have been many different form factor designs developed since the launch of IBM's original PC, some of which are shown below to illustrate the diversity of sizes available.
The original IBM PC motherboard used add-on cards which plugged into expansion slots for most of the constituent parts of the system. Many of these parts are now actually integrated in the motherboard itself. Whilst optimum performance on desktop PC's can still require the use of some add on cards, in my opinion this is not desirable for carputers, and the more functions built into the motherboard the better. This saves space and reduces power requirements.
The type of motherboard typically used in a carputer is sometimes termed a Small Form Factor (SFF) motherboard. However, SFF is not a rigid specification and you will see it used as "SFF PC" as well as "SFF motherboard", to describe a huge variety of equipment. The only requirement to fit the description is that it is smaller than the components normally used in desktop PC's.