How does StackOverflow's 'tags' textbox autocomplete work?
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.
You might also like this one:
Read the history here: http://code.google.com/p/jquery-autocomplete/wiki/History