A file manager is a character-mode or GUI interface to common CLI file and disk management tools, either directly, by serving as a front-end and calling on those tools, or functionally, by performing the same or similar operations internally. Historically, DOS's command line was rather poor and many users sought file managers for some relief. In UNIX, with its powerful and flexible command line, file managers didn't receive as much focus. Hence, most modern Linux file managers are modeled on DOS twin-panel file managers (for example, Krusader and Midnight Commander from Norton Commander - the classic twin-panel style); or they are modeled on Windows file managers (such as, Konqueror and Nautilus from Windows Explorer - the tree-style). While no substitute for the command line, they can make basic file management and certain tiresome repetitive operations more enjoyable.
File managers, or at least GUI-based ones, are also called file browsers. File browsers and web browsers share surprising large amounts of code, and file browsers and web browsers are largely interchangeable. (Try typing file:/// into the address bar of your web browser or file:///C:/ if your using windows)