To install, you have to create (or edit) a file in your userspace. For most people, this is Special:Mypage/monobook.js.
(People using the classic skin should edit Special:Mypage/standard.js instead, and people using other skins probably know which file to edit.)
Edit that page and paste in this line:
which expands to
Clear your browser's cache and when you try searching for an article, the drop down list should appear under the search input box.
The code is available under GPL at User:Zocky/AutoComplete.js.