Twitter has started rolling out its "Verified Account" program, which is supposed to tell users that a Twitter account is approved by the person connected to the account. You can tell if an account is verified by a checkmark and the words 'Verified Account' at the top of the account's profile page.
Twitter HQ hastens to add that if an account isn't verified it doesn't mean that it's fake, and also that verification doesn't mean the celebrity is doing the actual tweeting. It just means the celeb approves of the account. The program is still small and being handled on a case-by-case basis; the only way to request that your account be verified is by using a Feedback Form (which isn't live yet). It sounds like, for now, Twitter is limiting this problem to celebrities who have genuine problems with people impersonating them. One could guess that they scrambled to pull this together after in the face of mounting criticisms (and lawsuits) from outraged celebs.
There is no indication that any kind of fee is associated with getting a verified account at this time. I would expect that to change if they broaden the program to businesses and us regular folk.
You can find all the details here.