An applet actually specifies any small computer program designed to work with a very specific user interface. Usually this interface involves a display (plain canvas or more advanced windowing toolkit) and method of user input (mouse clicks, menus, keyboard, etc.) The point being, of course, is to use that specific interface to integrate applets into a larger program. Java applets work with standard Java UI components, like AWT or Swing, to integrate with your web browser.
Other examples of applets include Gnome applets, WindowMaker (GnuStep) dockapps, some Winamp plugins, even the system tray items in Windows. Of these, only WM dockapps are capable of running while not nested in their native program.
While it's expected that an applet will be a thin program, there are generally no real-life limitations on applet size (actually, Java puts a limit on the heap size for applets, but this goes for all Java programs, not just applets) or system priority. Some applet interfaces limit what kinds of actions the applet can perform; for example, Java applets (unless they go through a trust-gaining mechanism) can't make new IP sockets or open files, for obvious security reasons. Gnome and WM dockapps do not have such limitations, since they aren't automatically downloaded and executed by browsers.