Google announced this morning that they have made changes to the search box you see within the Google search results under a search results snippet. There are three changes:
(1) The box is now above the sitelinks of the search result.
(2) It offers auto-complete as you type, specific to your site's content.
(3) You can enable it to link to your site's internal search, as opposed to going back to a set of Google search results page limited by the site command.
Here is a picture:
Using schema.org Actions markup, you can communicate to Google where your site site search is located and Google will point the search form there instead of at Google.
Google has a lot of documentation on this over there.
Why not have the search results load on your page so you get the page impressions?
Forum discussion at Google+.
