See D. Butler, Lord Mountbatten: The Last Viceroy (1986); P. Ziegler, Mountbatten (1986); A. von Tunzelmann, Indian Summer: The Secret History of the End of an Empire (2007).
It differs from the official name of the British Royal Family or Royal House, which remains Windsor. The adoption of this surname does not apply to members of the Royal Family who are not descended from The Queen (her cousins, for example, and the descendants of the late Princess Margaret). The Order specifically applies the surname to those descendants of the Queen not holding Royal styles and titles but it has been applied to or informally used by members of the Royal Family descended from Queen Elizabeth II as their surname, as shown at the marriages of the Duke of York and the Princess Royal, both having been registered with Mountbatten-Windsor in their entries in the marriage registers.
The Mountbatten surname derives from the German town of Battenberg, in Hesse. Prince Louis of Battenberg changed his surname to Mountbatten (its literal English translation) during the First World War at the request of King George V. When then-Prince Philip of Greece and Denmark took British citizenship, he used this surname since he descends from the Mountbatten family through his mother, Princess Alice of Battenberg. It also might be seen as honouring the British Admiral of the Fleet Earl Mountbatten of Burma, the Prince’s uncle.
None of the above actually officially hold the surname as part of their legal name, owing to the styles that are used for members of the Royal Family, and to the ambiguous wording of the proclamation. For example, when the Duke of York was in the Navy, he was referred to as Lieutenant His Royal Highness, The Prince Andrew before he became The Duke of York, and Lieutenant His Royal Highness, The Duke of York afterwards - but not Lieutenant Mountbatten-Windsor. While Mountbatten-Windsor was entered into the marriage register for Prince Andrew and Princess Anne, the Prince of Wales was entered as simply "The Prince Charles Philip Arthur George".
It has been reported in the American press that as a practical day-to-day matter in their active Royal Navy service, both Charles and Andrew were referred to "Lieutenant Windsor". A 10-syllable name simply isn't a practical one for everyday use.
Both Princes William and Harry have used "Wales" as a last name during their schooling. Both were known as Officer Cadet Wales at the Sandhurst Military Academy. They wrote the name "Wales" on their socks and underwear for laundering. Princess Beatrice and Princess Eugenie of York both similarly use "York" as a last name.
The Earl of Wessex has styled himself "Edward Wessex" for his television series Crown and Country since acquiring that title upon his marriage. Prior thereto, the show's credits listed him as "Edward Windsor."
Seemingly the only people who would officially hold the surname under the Order-in-Council would be any male-line great-grandchildren of the Queen not in direct line to the throne, i.e. the children of any sons of the Duke of York and Earl of Wessex. Similarly, in the event that any male-line granddaughter of the Queen were to have a child whilst unwed. Also, when (and if) Charles becomes king, any male-line-great-grandchildren that he would have who were not in direct line to the throne would also use the surname. The surname applies to any descendants of The Queen and Prince Philip who do not hold the style of Royal Highness and rank of Prince/Princess of the UK.