What is a "File System" in a Computer?
A file system (or filesystem, abbreviated FS) is a method to organize data in a computer as "files." Each "file" has a name (a filename) that identifies the data that's associated with it. Typically, files can be placed inside "folders" to organize them, and folders can also be placed inside folders, becoming subfolders. This creates a hierarchical filesystem. To identify a file or folder in this filesystem, we use a text code called the filepath.

Although the typical computer will have a filesystem, computers don't come with filesystems in them by default. A filesystem is something you have to "install." More specifically, the data in the computer is permanently stored in a mass storage device like a hard disk or SSD. These devices don't understand the concept of files, only of data as bytes. The device needs to be split into partitions, and the partitions formatted to house a filesystem in them. When we talking about "formatting the computer," we're talking about installing a filesystem like FAT, NTFS, Ext4, or Brtfs in a partition of a mass storage device that's connected to the motherboard of the computer.
