Twitter is now trialling verifiable accounts for popular Twitterers (i.e., in beta). This service is being tested on well known folk like artists, athletes, musicians, actors, etc. to prove they are who they say they are. This does not mean, however, that they actually tweet but that the account is legitimately connected with them in some way.
The approach taken by Twitter is to add a new badge to the profile sidebar to clearly demarcate that an account has been verified (see the screenshot below of Pete Cashmore's @mashable account as an example).
Right now these verifications are being tested on a small of set of people to see how the community reacts and to see if verified account badges are trustworthy and workable from both parties.
This means that if you are an an ordinary user like me, verified accounts may never be an option or even necessary.
Read this Twitter help page for more.