Definition: Computers use particular kinds of file systems to store and organize data on media, such as a hard drive, the CDs, DVDs, and BDs in an optical drive or on a flash drive. Any place that a PC stores data is employing the use of some type of file system.
A file system can be thought of as an index or database containing the physical location of every piece of data on a hard drive.
The Microsoft Windows operating systems have always supported, and still do support, various versions of the File Allocation Table (FAT) file system. In addition to FAT, all Microsoft Windows operating systems since Windows NT support a newer file system called New Technology File System (NTFS).
Some other operating systems also take advantage of FAT and NTFS but many different kinds of file systems exist. Wikipedia has a comprehensive list of file systems if you're more interested in the topic.
Examples: "The hard drive that I have the Microsoft Windows 7 operating system installed on uses the NTFS file system while my friend's Mac OS X computer uses the HFS Plus file system."