Google announced that it's now showing a new and improved sitelinks search box within search results, which will make it easier to find specific content on third-party websites from Google itself.
The box is more prominent, and supports autocomplete. Here's what it looks like for YouTube:
You can mark up your own site so that Google has the ability to display a similar functionality for your content. Google explains:
You need to have a working site-specific search engine for your site. If you already have one, you can let us know by marking up your homepage as a schema.org/WebSite entity with the potentialAction property of theschema.org/SearchAction markup. You can use JSON-LD, microdata, or RDFa to do this; check out the full implementation details on our developer site.
If you implement the markup on your site, users will have the ability to jump directly from the sitelinks search box to your site’s search results page. If we don’t find any markup, we’ll show them a Google search results page for the corresponding site: query, as we’ve done until now.
More on the markup can be found on Google's Developers site.
Image via Google