Occasionally you might see a Google search result ( a SERP) with handy links, even a search input box. They make it easier for searchers to find relevant content in your site,and so are very likely to attract more visitors to your site.

How can you get this extended SERP showing on your site ?

Sadly, as mere mortals we don’t have the option to ask for these extended SERPs (Google calls them Sitelinks)
Google decides if it will display Sitelinks with the SERP, but it seem to be based on the site home page having a Google PageRank of 5 or more and the search keyword very relevant to the site

For example the Medibank Private site has a PageRank is 7 and a search of medibank private is as you could imagine very relevant hence a Google search result displays:

It is possible to use Google WebMasters Tools to remove selected Sitelinks

So if your site has a PageRank of 5+ then highly relevant searches may bring up a SERP with Sitelinks