The motherboard or mainboard is the component that connects all the different parts of your computer. Often it has some integrated components like a sound card, enabling your computer to play sound, or a network card, to connect your computer to the network. The standard form factor (size) for motherboards is ATX, you need at least a midi-tower to fit one of those in. Micro-ATX motherboards are often more complete and have more onboard devices, like a video board and fit in smaller cases, but are generally less expandable.
