The blue badge is now finally within your reach—Twitter has created an official process to have your account verified, proving once and for all that you are who you say you are on the site. It's not a sure thing, but at least there's a way to get your request in front of someone without going through back channels.
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.