Many Twitter users, including celebrities, companies, brands, government agencies, and more, have a desire to prove that their Twitter accounts are legitimate. For example, many celebrities have Twitter accounts to interact with their friends and fans. At the same time, there are many Twitter accounts created by fans of those celebrities or people posing as those celebrities for their own entertainment. In an effort to prevent identity confusion and distinguish between Twitter accounts maintained by the "real" person behind the name or persona vs. fan accounts and so on, Twitter created the Verified Account system and badge, which you can see in action on Twitter's own Twitter profile here.
Note that the Twitter Verified Account system is still in its beta form, which means it hasn't rolled out to the entire Twitter user base yet.
