What does it mean if an application is AppleScript "attachable"?

It means that the application allows you to write custom AppleScripts and access the scripts easily within the application, usually by way of a menubar icon and contextual menus. A common example would be Outlook Express and Tex-Edit Plus. Both of these have menus that contain scripts, and you can add your own scripts by simply saving compiled AppleScripts in a specified folder.

To take this a step further, check out Leonard Rosenthol’s terrific OSA Menu (OS 9 and back) which puts an AppleScript icon in your menu bar for every application you are in!; and Red Sweater Software’s FastScripts, which is OSX’s replica. With less powerful features you can use Apple’s built-in Script Menu, located within your /Applications/AppleScript folder.