<p>I know they're using a jQuery plugin, but I can't seem to find which one they used. In particular, what I'm looking for is autocomplete with exactly the same functionality as SO's autocomplete, where it will perform an AJAX command with each new word typed in and allow you to select one from a dropdown.</p>

Note that the tag editor
has been completely re-written now
, and no longer resembles the original, simple text box w/ suggestion drop-down that adorned the site for nearly three years.
If you're interested in the new form, see this Meta question:
https://meta.stackexchange.com/questions/102510/can-i-use-the-tag-textbox-script
Autocomplete
is the plugin used originally, albeit with various tweaks and customizations made to it over the years.