"By suggesting more refined searches up front, Google Suggest can make your searches more convenient and efficient by keeping you from having to reformulate your query. Google Suggest might offer suggestions that you will find novel or intriguing," explains Google in an interesting FAQ.
The feature is enabled by default, but you can disable it in the "Settings". For now, YouTube Suggest seems to be live only for international sites like YouTube UK and only if you search from the homepage, but it should be available at YouTube's main site in the near future.
To get the suggestions, YouTube uses this simple JSON call:
http://suggestqueries.google.com/complete/search? hl=en&ds=yt&json=t&jsonp=callbackfunction&q=QUERY
where ds=yt defines the search's scope (YouTube), while q=QUERY includes the characters typed by the user. A similar URL is used by Google News to suggest news sources in advanced search, so we can expect an API for query suggestions:
http://news.google.com/complete/search? hl=en&ds=ns&js=true&q=QUERY
This URL works as well:
http://suggestqueries.google.com/complete/search? hl=en&ds=ns&json=t&jsonp=callbackfunction&q=QUERY
YouTube constantly experiments with new features and most of them are related to the way people navigate the site or discover new videos. A recent experiment added a search box below the list of related videos so that people can search and see the search results while watching a video. The only problem was that you couldn't add the results to the Quicklist in order to build dynamic playlists.
Update (May 16): YouTube Suggest is now live at youtube.com.