Age of your website, pages your visitors visit, sitemap.
A.k.a. Content, content, content.
They are called "Sitelinks". Google will pick them automatically and there is no way you can add them manually. You can disable the ones you don't like in Webmaster Tools.
I think the main advantage is that people is more likely to visit your site because it takes up more space and stands out more in the search results. I don't know exactly when they are shown but I think it's only for the first result and mostly when searching a brand or a website name.
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