What is a verified account? – Any account with a blue verified badge on their Twitter profile is a verified account.
Why does Twitter verify accounts? -Verification is currently used to establish authenticity of identities of key individuals and brands on Twitter.

First things First

Determine if you qualify for a verified Twitter account. Twitter verifies as few accounts as possible and only for legitimate reasons.

• Reasons for verification include being a highly recognizable public figure (musicians, actors, athletes, artists, public officials, public or government agencies, etc.), or if your name and likeness is parodied or impersonated on multiple Twitter accounts, leading to identity confusion.
• Twitter will not consider you for verification based on your number of followers. Twitter urges Tweeters to “Please note that follower count is not a factor in determining whether an account meets our criteria for verification.” Similarly, the number of tweets you post irrelevant.
• For more information, read the Verified Account terms. These terms explain what a verified account is, what it means to be verified, who has the verified badge, identifying a verified account, etc. They can be found here.

Ask nicely. Although Twitter do not officially accept requests for verification, you may be able to speed up the verification process by sending a message to them directly (but only if you meet the criteria outlined above). Send a direct message to this address, making sure to include the following information:

• Account name
• Full name
• Location
• Official website (you will be given a suggestion to put the Twitter logo or badge on your official website to speed up verification)
• Bio information (information about yourself in fewer than 160 characters)
• Primary contact name (referring to the individual who manages the account)
• Additional Contact Information

Wait for Twitter to respond. If you don’t get a reply shortly, you can contact Twitter by snail mail or through Twitter. Keep in mind that Twitter deals with a high volume of verification requests, and that it focuses first on the most “highly sought users. Follow Twitter’s instructions for verifying your account. If Twitter decides that you qualify for a verified account, they will reach out to you via direct message. Click the link in the direct message to be taken to finish the process. The final part of the process has 3 parts:
(1) Learn how to Tweet effectively.
(2) Connect with other Interesting Twitter Users.
(3) Protect your Account.

1. Learn how to Tweet effectively gives you a choice between 2 tweets and asks you to choose which one is better. It has the form of a quiz, but there will be no negative effects if you do not answer correctly.
2. Connect with other Interesting Twitter Users gives you the option to follow other verified accounts. Twitter believes that this gives you more legitimacy as a verified user.
3. Protect your Account asks you to enter a phone number that Twitter can call if there are any problems with your account. Once you have completed this step, your account will be verified.

Do verified accounts have access to extra features? Yes, verified account holders have access to the following extra features:

• Filters in the Notifications page allow verified account holders to display Interactions and Mentions in one of four ways: All (default), Filtered, Verified accounts you follow, People you follow.
• Visitors to verified account profile pages can select between two timeline options: No replies or All. No replies, which is the default setting, displays Tweets that are not direct @ replies to fans or followers. All displays every Tweet, including @ replies.
• Verified account holders can choose to receive DMs from all followers by opting in from the Account settings tab.
• Verified account holders have access to account analytics, including data and characteristics about Tweet engagement and followers.
• Get Instagram Verified Badge
• Get Verified on Facebook: Page or Profile

Another way: 

You can login to your Twitter and submit this form https://support.twitter.com/articles/20174631

To be able to complete the form, your account must have the following:

• A verified phone number

• A confirmed email address 

• Note: If the account is a company or organization account, the email address associated with the account should be a company or organization email address.

• A bio

• A profile photo

• A header photo

• A birthday (for accounts that are not company, brand, or organization accounts)

• Note: This information will be displayed to the audience that you’ve chosen. There are visibility settings available for your birthday that allow you to separately control who on Twitter can see your birth year and who can see your birth day and month in your Twitter profile. Read more here.

• A website
• Tweets set as public in Tweet privacy settings