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Re:Ask HN: How does Google select text for header/subheader ... by stephen_dryden on Thursday June 30, 2011 @11:32AM (#2714430)
It used to be the case that the header and sub-header's were created from <h1> and <h2> tags on your website. That is why it is advised to only have one <h1> per page. I'm unsure whether or not that is still the case. Google are constantly changing how they display and rank results.

I would advise doing some research into search engine optimisation(SEO).

Re:Ask HN: How does Google select text for header/subheader ... by mooism2 on Thursday June 30, 2011 @11:42AM (#2714463)
The link text is usually from the page's <title> element. The text in black following is sometimes taken from the page's <meta name=description>, and sometimes from text on the page Google feels is relevant.

I think that is what you mean by "header" and "subheader"; if not, please clarify.