The application menu is a menu on all versions of the Mac OS starting from System 4.2 (as part of the Switcher utility through to the last version of the Classic Mac OS, Mac OS 9.2.2. Strictly speaking, it is no longer part of Mac OS X.
Its existance allowed the user to switch between different application programmes that were open and running.
The application menu became fully independent as of System 7. As of Mac OS 8.5, it was even resizable, so that the name of the application was shown. Dragging the menu off the menu bar creates the Application Switcher.
A variant of it (strictly speaking, not the real application menu) exists under Mac OS X. The menu next to the Mac OS X Apple menu shows the name of the active application. However, it on its own cannot change between different applications. The third-party utility X-Assist brings the Mac OS 8.5 variant back to Mac OS X.