FAT12: The oldest type of FAT uses a 12-bit binary number to hold the cluster number. A volume formatted using FAT12 can hold a maximum of 4,086 clusters, which is 2^12 minus a few values (to allow for reserved values to be used in the FAT). FAT12 is therefore most suitable for very small volumes, and is used on floppy disks and hard disk partitions smaller than about 16 MB (the latter being rare today.)
FAT16: The FAT used for most older systems, and for small partitions on modern systems, uses a 16-bit binary number to hold cluster numbers. When you see someone refer to a "FAT" volume generically, they are usually referring to FAT16, because it is the de facto standard for hard disks, even with FAT32 now more popular than FAT16. A volume using FAT16 can hold a maximum of 65,526 clusters, which is 2^16 less a few values (again for reserved values in the FAT). FAT16 is used for hard disk volumes ranging in size from 16 MB to 2,048 MB. VFAT is a variant of FAT16.
FAT32: The newest FAT type, FAT32 is supported by newer versions of Windows, including Windows 95's OEM SR2 release, as well as Windows 98, Windows ME and Windows 2000. FAT32 uses a 28-bit binary cluster number--not 32, because 4 of the 32 bits are "reserved". 28 bits is still enough to permit ridiculously huge volumes--FAT32 can theoretically handle volumes with over 268 million clusters, and will support (theoretically) drives up to 2 TB in size. However to do this the size of the FAT grows very large; see here for details on FAT32's limitations.