Software

In short:

Java App, is a compiled program meant to run within a browser or independantly on any platform that has Java installed.

Java Script, is a relatively shorter uncompiled bit of scripting that is compiled on and by the user's browser at the time it is needed.

In less short:

Java Script is uncompiled java code included in html code to perform some function. It runs on the client side within the browsers java sandbox.

One use is having java script confirm that form fields are filled in with valid entries.

Java Applets are compiled binaries distributed as there own program which requires Java as it's programming basis. These may or may not be used within a browser and may be used outside of a browser.

One example is Limewire. On Windows, you'll notice that it requires the Java runtime environment. On Linux based OS (Ubuntu, Mandriva), it also requires the Java runtime environment. The result is the same binary blob can be run on any software platform that will run Java.

One of the other's with more programming background can probably add more too the discussion.