Since the marriage of Princess William and Kate Middleton, Prince and Princess William of Wales have been given a royal dukedom by The Queen and are to be known as the Duke and Duchess of Cambridge.
Catherine will be known formally as Her Royal Highness the Duchess of Cambridge. Her other name, as the wife of Prince William of Wales, the Duke of Cambridge, will be Princess William.
She was not born a princess so she is not (yet) Princess Catherine.
When the current Prince of Wales (Prince Charles) ascends the throne upon the death of The Queen and becomes King Charles III (or perhaps King George VII), the Duke and Duchess of Cambridge will become the Prince and Princess of Wales as well as Duke and Duchess of Cornwall and Lord and Lady of the Isles... at this point Catherine will become HRH Princess Catherine of Wales.