A disk file system is a way to store files on a disk, USB, DVD, netowrk, etc. There are many different file system types and different types are used by different operating systems. If you get on a Linux system and use the l option you will see that Linux can provide many different options including Microsoft formats.
It is important to understand that the features and purpose of each file system could be completely different. For example the file system you would want to use on a Flash disk could be very different from what you would want on a hard disk because the Flash disk has limited writes.
Currently Microsoft is using NTFS, New Technology file System, which is designed to enhance performance, security, reliability and disk space utilization over FAT, the file system used before. It also provides ACLS or access control lists to allow different users rights to the same file.
Linux is currently moving toward ext4 which is a file system that is designed to be a journaling system to help in rebuilding the system if there is a crash. It also has performance enhancements and the ability for much larger volumes. Here is an article on ext4:
http://beginlinux.com/server_training/8-ubuntuadmin/1255-crash-testing-ext4-on-ubuntu
In a nutshell, file systems are ways that operating systems store files, the exact file system of choice often changes as file systems store more data and are choices by the OS.