So Twitter is a great tool to know what is going on in just about any field of interest that you may have. But a particular sphere of culture and society that requires some delicate care in the nature of communications, is politics. We can only imagine how many Bill Clintons and Sara Palins there are out there in Twitter world, and fortunately the powers of the enterprise have managed to implicate a filtering process to protect those that are often victims of impersonation. The blue verification check mark has been applied if a Twitter executive has been in direct communication with the owner of the account and has been given sufficient proof (Ie: a link to their Twitter account via official website, etc) to brand the account as Verified.
This verification process is still being tested and can only be done by request through Twitter's feedback form. A request for business verification is certainly no promise, as Twitter has specified that businesses are not included in the Verification trials that are currently under way.
As you can imagine, politicians are actively using Twitter during campaigning. Barack Obama's page is my current fave, check it out here: http://twitter.com/BarackObama, and yes... it is verified!