You can send requests now to have your Twitter account verified

On Tuesday, July 19th, Twitter announced that it has created an online application process (click here to start) for Twitter accounts to receive verified status.
Verified accounts on Twitter allow people to identify key individuals and organizations on Twitter as authentic, and are denoted by a blue badge icon. An account may be verified if it is determined to be of public interest. Typically this includes accounts maintained by public figures and organizations in music, TV, film, fashion, government, politics, religion, journalism, media, sports, business, and other key interest areas.
AndroidPolice notes that:
There are several account requirements before you can even submit a request. For example, you need to have a profile picture, a verified phone number, a verified email, a birth date (for personal accounts), and so on. When submitting a request, you'll need to supply at least two links that help verify your identity and noteworthiness. You will also be asked to upload a photo ID (Twitter says it may ask for this, but it's on the form).
After the application is submitted, Twitter will reach out via email to respond. Presumably you have to be at least somewhat of a known quantity on the internet to get the blue badge. If you are denied, you can try again 30 days later.