The portion of a filename, following the final point, which indicates the kind of data stored in the file - the file type.
Many operating systems use filename extensions, e.g. Unix, VMS, MS-DOS, Microsoft Windows. They are usually from one to three letters (some sad old OSes support no more than three). Examples include "c" for C source code, "ps" for PostScript, "txt" for arbitrary text.
NEXTSTEP and its descendants also use extensions on directories for a similar purpose.
Apart from informing the user what type of content the file holds, filename extensions are typically used to decide which program to launch when a file is "run", e.g. by double-clicking it in a GUI file browser. They are also used by Unix's make to determine how to build one kind of file from another.
Compare: MIME type.
Tony Warr's comprehensive list.