The text snippet circled in red might be from the full text of the web page, from a description from ODP (see this example), or, as in this case, it might come from the meta description tag like the one I have in my web page:
<meta name="description" content="Frank McCown, Assistant Professor of Computer Science">Just for fun, I changed this tag a few weeks ago. Although Live has re-crawled my page since then, they have not re-indexed the meta tag. But Google (and Yahoo) have:
This of course is a very misleading description of my web page, but Google doesn't appear to care. A Google search for this description will not make my page appear in the SERP, so the misleading description is only hurting me (and those trying to find me).
I'm a little surprised Google doesn't at least search for the terms from the description in the accompanying web page just to make sure the description is not misleading. I suppose it's better to err on the side of giving your web page author the benefit of the doubt than to deal with all the complaints of those whose tags might be ignored.
Matt Cutts from Google discusses text snippets on YouTube.