When it comes to British royalty, there are two ways to become a princess: you either need to be born the daughter of a prince (or the British sovereign), or you have to marry one.
That second option is relevant, for example, to Catherine, Duchess of Cambridge. There is a lot of debate in the media over whether or not to call her a princess, but simply put, Kate is a princess, but she's certainly not Princess Kate.
On her wedding day, Kate Middleton took on her new husband Prince William's title, effectively becoming Princess William of Wales, in addition to Her Royal Highness, Duchess of Cambridge.
Her princess status was doubly confirmed with the arrival of her son, Prince George. His birth certificate revealed that her official occupation is Princess of the United Kingdom. Not too shabby a job.
In much the same way, grandchildren born to the sons of the monarch receive the title of prince or princess, but those born to the daughters of a monarch don't. In the case of the family of Prince and Princess William of Wales: Prince George's children will automatically be princes and princesses; Princess Charlotte's kids will not (unless, of course, a tragedy upends the British line to the throne).
According to Lucy Hume, associate director of Debrett's, which is the trusted resource when it comes to matters of British peerage, "Royal titles are inherited through sons, so if Princess Charlotte has children they would not automatically inherit the titles 'HRH,' 'Prince,' or 'Princess.'"
This is also why Beatrice and Eugenie, the daughters of Prince Andrew, are princesses, but Zara Phillips, the daughter of Princess Anne, is not.
You can, if you wish, opt out of the whole thing. Prince Edward's daughter Louise is a Lady, not a princess, a peerage decision that was made for his unborn children when he married Sophie Rhys-Jones. According to the BBC, the decision reflected, "the clear personal wish of Prince Edward and Miss Rhys-Jones as being appropriate to the likely future circumstances of their children."
How very modern of them.
That doesn't mean Louise and Zara are unencumbered by royal pomp. "The monarch may offer to bestow a royal title upon his or her daughter's children," says Hume. "For Peter and Zara Phillips, the Queen offered to give them a royal title when they were born, but Princess Anne and Captain Phillips opted to decline this offer."
This question of titular inequity has only really become a concern in the 20th century. For the bulk of British history, marriage was a transactional means to fortify alliances between nations, so princesses typically married princes and their children took on the titles of their father's country. Now that love matches are standard protocol, the question of titles comes into play.