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Well, not different kinds of applescripts, but different ways to execute and handle applescript code.
We can use compiled scripts. We can open these in any script editor and run them choosing the "Run" button/menu. We can also execute these from any script menu, such as Apple's Script Menu or the one provided by some applications: Entourage, BBEdit, etc.
As standalone applications (double-clickable applications, just as GraphicConverter or Dreamweaver), we can save our scripts as applets or droplets. Droplets are the same as applets, but they can accept and work with files/folders opened via drag & drop. They will respond to the "open" apple event.
These are the most common ways to run applescripts. But there are lots of variations. We can run uncompiled applescript code on-the-fly using the do script command provided by some applications, such as Smile or Tex-Edit Plus. We can also use embedded scripts in some applications or languages. For example, we can use applescript code within FileMaker's ScriptMaker scripts, use applescript code in shell scripts using the osascript command or use applescript as agent in a CGI script.
In newer OS X versions we have "packages" (the Cocoa look & feel), and the new formats ".scptd" (package containing various compiled scripts) and ".app" (Cocoa-style application package).