Google Allows Developers to Integrate a Search within a Site Feature
As of September 5th, 2014 Google announced a minor change to the search box you find within the results of your initial search. Usually, when you search for a brand name for example, Google would add a number of search results below the initial search. This feature allows you to search around that specific site only.
The new results are very similar to this, with a few minor changes. The changes include moving the search bar above the sitelinks in the snippet and giving developers the option to tell Google to direct the search results to their website.
If you want the searches from the snippet to go directly to your website, you can do this by adding some schema to your website. For more information on this change, click here.