There are lots of different motherboard types. Laptop motherboards are normally purpose built for the laptop where as desktop motherboards usually follow a standard. There are over 40 different types of desktop motherboard (link) but the major desktop motherboards are ATX and ITX.
ATX stands for Advanced Technology Extended. This motherboard was standardised in 1995 by Intel.
It contains either a 20pin (older) or 24pin (newer) power connector. The standard ATX motherboard is large in size so can accomodate more expansion slots (eg PCIe)
This motherboard is the same as the ATX except reduces its size by limiting the number of expansion slots. It has similar mounting points and power sockets meaning it can be placed in a case designed for a standard-atx motherboard.
The ITX motherboard types were created in 2001 by VIA technologies as the ATX motherboards weren’t small enough for new technology. These motherboards require lower power and have a limited number of expansion slots or no expansion slots. It uses the same mounting points as the ATX so can be installed in a case designed for an ATX motherboard.