File systems and writing one
The MBR refers to the format used by (many but not all, and slowly being replaced by GPT) disk to identity partitions. A partition is simply a contiguous range of blocks on the disk. A partition is often used to store data, in the form of files and directories. But of course a disk knows nothing about files and directories. A file system is a system that controls how your data is stored on the disk. A file system is really two things:


it specifies how the data is organized and accessed (directories, files, links, permissions, encryption, ...): this is the logical aspect
it specifies how this data is encoded on the disk: this is physical aspect


There exists many file systems: FAT, exFAT, NTFS, ext, btrfs, HFS, XFS,  ISO 9660 and others. FAT is a file system invented by Microsoft. What this means is that they designed the logical and physical aspects of it, and they applied and were granted a few patents on it. They also wrote the Windows driver for FAT, but note that anyone can do that (assuming the file system is properly documented and ignoring royalties issues) and indeed linux and MacOS have their own driver which follows Windows's specification.

Now as I explained a file system specifies how the data is encoded on the disk and FAT is not exception. The first sector of a FAT partition (also known as the boot sector, or super block) is what is described in the link you gave: is starts with 3 bytes for a jump instruction and is followed by the BIOS/OEM Parameter Block (which has a different format on FAT12, FAT16 and FAT32). This OEM block is specific to FAT and does not exist on other file systems. Other file systems use a different format. I might add that calling this the OEM/BIOS block was very unfortunate because it really has little to do with BIOS or OEM, but that was the name Microsoft came up with. It should really be called the FAT Parameter Block.

To finish, designing a good file system is hard because they are many trade-offs to make. FAT is simple and relatively robust, but sucks at about everything else if I may say. Sadly it is also the only file system that all OSes support and so is the de facto standard for interoperability.