In order to understand the syntax of file names, you need to understand how the file system is organized into a hierarchy of directories.
A directory is a file that contains information to associate other files with names; these associations are called links or directory entries. Sometimes, people speak of ``files in a directory'', but in reality, a directory only contains pointers to files, not the files themselves.
The name of a file contained in a directory entry is called a file name component. In general, a file name consists of a sequence of one or more such components, separated by the slash character (`/'). A file name which is just one component names a file with respect to its directory. A file name with multiple components names a directory, and then a file in that directory, and so on.
Some other documents, such as the POSIX standard, use the term pathname for what we call a file name, and either filename or pathname component for what this manual calls a file name component. We don't use this terminology because a ``path'' is something completely different (a list of directories to search), and we think that ``pathname'' used for something else will confuse users. We always use ``file name'' and ``file name component'' (or sometimes just ``component'', where the context is obvious) in GNU documentation.
You can find more detailed information about operations on directories in File System Interface.