Google announced a few changes to the search box you find within the Google search results.
Often, when you search for a brand name, Google will add a search box within the search snippet result for that site. That box allows you to search within that specific site only.
The results are similar to the site command Google allows users to use when searching for a specific domain only.
A few updates include:
(1) Moving the search box above the site links in the snippet
(2) Adding autocomplete results to the search results in that box and
(3) Enabled developers to instruct Google to send the search results directly to their own site.
The search box is now more prominent which supports autocomplete and with the use of the right markup you can send your user directly to your website’s own search pages.
Here is a picture of the new interface:
If you want to jump the searchers using that box directly to your site search results page, you can do so by adding some schema to your site. If there is no markup then the google search result page for the corresponding website is shown.