Those extra links you see under a website’s listing in the Google search results are called Sitelinks (see Google’s description).
There is nothing a website can do to get them, though having good website navigation can help Google provide them. The Sitelinks are given to a site by an algorithm which is why it is next to impossible to influence that. You can, however, remove Sitelinks that you do not want via your Google Webmaster Tools. In your Webmaster Tools you can see what Sitelinks Google have for your site. Even if your site ranks at number one, does not necessarily mean that the Sitelinks will be displayed in the search results.
The search algorithm displays them in the search results when it is confident that the searcher is using a search keyword(s) or phrase that means that they are probably looking for the website in question.
For example, if you search for ‘Apple Inc”, then its probably pretty obvious that you are looking for the Apple corporate website and you will get Sitelinks under the Apple Inc listing:
However, if you search for ‘Buy Apple computer’, then it is obvious to the search algorithm that you are not searching for any specific website, so there will generally be no Sitelinks in the search results.