Google's Webmaster Central Blog recently announced that the search engine was enhancing its secondary search box for site-specific results. As the Webmaster blog explains, the "sitelinks search box" is meant to make it easier for users to find a specific page on an organization's website.
For example, a search query for "youtube" returns a list of results as well as another search bar labeled "Search youtube.com" with which users can perform a site specific search — all without leaving the first page of results. Google is gradually going to move the sitelinks search box to a spot directly beneath the main search result, making it easy for users to refine their searches.
To use the sitelinks box, a site must have a working site-specific search engine and be marked up in the proper way. Google's Developer site has more details about the markup requirements and you can read more about the enhanced sitelinks search box by visiting the Google Webmaster Central Blog.