The ability to easily carry around and customize the display of a great many texts is the core thing, but that would apply to any decently-designed dedicated e-reading device anyway.
So, as for Kindle-specific things that I really like:
The ability to highlight text and make notes, and have them exported so I can use them elsewhere.
Search and dictionary lookup are also contenders, but it would be nice if they allowed you to save particular searches so you could easily re-visit the results later without having to run the search again and again. Also, Boolean search or regexp.
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