Princess Beatrice is the oldest child of Prince Andrew and Sarah Ferguson, the Duke and Duchess of York. She is a blood princess and a direct descendant of the reigning monarch, Queen Elizabeth.
Prince Andrew became the Duke of York in 1986, on the very same day he married Sarah, who is more popularly known as Fergie. By virtue of their marriage, Fergie also received the title of the Duchess of York but their first-born child Princess Beatrice will never get to inherit the title or the dukedom from her parents.
According to Daily Express, dukedoms in the British royal family will only apply to the male descendants of the monarch. While a blood princess, Beatrice cannot inherit a title unless she is married to a titled royal as well or the Queen grants her husband a new title, which would be rare.
This rule on titles and nobility dates back to centuries. So, when the Duke and Duchess of York die, the peerage will revert back to the Crown. It will remain "extinct" until the next monarch decides to grant this to another male heir.
Once the dukedom is granted, the royals who carry the titles will have these for life. Hence, even after the Duke and Duchess of York divorced in the 1990s, Fergie is still sometimes called by her title.
However, Fergie's royal name is no longer styled as "Her Royal Highness the Duchess of York" since she is not a member of royal family anymore. Fergie may continue to use her title until she remarries.
But, outside of the British royal family, Princess Beatrice could have a different title because of her marriage to Edoardo Mapelli Mozzi in July 2020. Edo is the only male heir of Count Alessandro Mapelli Mozzi, who is from an aristocratic family in Italy. This makes Edo a count and his wife a countess, automatically.
According to Edo's father, the children of Princess Beatrice and Edo will also be counts and countesses. In Italy, however, countesses are also called contessa or nobile donna (noble woman) so Princess Beatrice may use any of these titles.
But following her marriage, Princess Beatrice officially dropped her York title just like when her younger sister Princess Eugenie married Jack Brooksbank in 2018. Both princess no longer use "of York" next to their names in official royal documents. Instead, they are more formally called as Mrs. Edoardo Mapelli Mozzi or Mrs. Jack Brooksbank. They may also still be called princesses since these are the titles they've earned at birth.