What is a Filesystem

A filesystem is like the book keeping of the files, their location on the disk, the free space that exists and many other stuff.
It's very important, unlike the partition table and the entries of partitions since if it gets corrupted the files can be difficult to retrieve.
There are many different filesystems, each with different features. For example NTFS FAT ext4 etc.

Partition table, Partitions, What Happens When a File is Deleted, Testdisk