Google Allows Developers Integrate With Search Within Site Feature
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 a specific domain only. The few updates include (1) moving the search box above the sitelinks in the snippet, (2) adding auto-complete 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.
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. Here is documentation on the Sitelinks Search.