How to customize the site in search engine results?

This is a really good question! However, the answer can be found relatively easily with a site:stackoverflow.com "A language-independent collaboratively" search.

The first thing you need to know:

https://support.google.com/webmasters/answer/35624?rd=1

We use a number of different sources for this information, including descriptive information in the title and meta tags for each page. We may also use publicly available information—for instance, anchor text or listings from the Open Directory Project (DMOZ)—or create rich snippets based on markup on the page.

If a description meta tag is not found for a page, Google will look for a snippet in content or in other places. Much of the time, it depends upon upon the search query. Since this is clearly a brand search for a known brand, it is likely that a different set of criteria are applied as opposed to a term based search.

This SERP snippet comes from DMOZ.

http://www.dmoz.org/search?q=stack+overflow

See item #2:

Stack Overflow - A language-independent collaboratively edited question and answer site for programmers.

I did a few other searches and checked JavaScript and looked for mark-up, however, nothing showed up. The only place I found a source for the SERP snippet was DMOZ.

For what it is worth, it seems this question has been asked on StackOverflow: https://stackoverflow.com/questions/3796793/how-to-programmatically-provide-site-structure-and-url-path-to-google-search

Your best bet are title and meta description tags. These will be used in most cases, although there can be exceptions.

Stack Overflow's description is most likely taken from DMOZ (http://www.dmoz.org/Reference/Ask_an_Expert/Computers_and_Technology/). Google will sometimes use the description from DMOZ instead of the provided meta description or a Google-generated description.

Unfortunately, this behavior can not be controlled by the webmaster. In this example, Google probably decided to use the DMOZ description since Stack Overflow provides no meta description and the page itself also does not offer enough content to generate an adequate description.