When Meghan Markle and Prince Harry introduced their son Archie to the world, many were surprised by his surname. Some were expecting him to have the surname Sussex, the format used by his cousins Prince George, Princess Charlotte and Prince Louis. Instead he has Mountbatten-Windsor, and while many were a little confused this was entirely expected. Kate and William's kids all have a HRH title, which means they technically don't really need a surname. But obviously there are some instances where they do need one, as HRH Prince George doesn't exactly roll off the tongue for the school register.
So they use Cambridge after their dad's Dukedom, which was given to him on his wedding day back in 2011. William and Harry followed the same system when they were growing up, using the surname Wales from their dad's title the Prince of Wales. However little Archie is a bit different as he doesn't get a title due to his position. The Queen could have given him one if she wished, but she didn't - a decision Meghan and Harry probably would have fully supported as part of their bid for him to have a 'normal' life. The lack of HRH means he needs a surname, so was given the Royal Family's official one - Mountbatten-Windsor.
However, the name was the outcome of years of upset in the family and rows between Philip and his in-laws. Before marrying the Queen, he was Prince Philip of Greece and Denmark of the House of Schleswig-Holstein-Sonderburg-Glücksburg. However, this wasn't neutral enough so he adopted the name Mountbatten after his grandparents. When the then Princess Elizabeth had the couple's first child Prince Charles, he assumed they would take on his name in the traditional way. But Prime Minister Winston Churchill didn't agree and wanted them to continue to use the name Windsor, which was the official Royal Family surname.
The Queen's grandmother Queen Mary agreed, and it caused such a row that the matter was even discussed in Parliament. In the end, the Queen made the decision to go with her family's views, and the name Windsor was used. At the time, Philip reportedly said, "I am nothing but a bloody amoeba. I am the only man in the country not allowed to give his name to his own children."
However, when the couple fell pregnant with their third child Elizabeth was sitting on the throne, which changed things slightly. Philip reportedly refused to drop the issue, and in 1960 she went to see the Harold Macmillan, who had then taken over as PM.
She said, "She absolutely needed to revisit" the issue and admitted it "had been irritating her husband since 1952". Finally, a comprise was met, and on February 8, 1960 - 11 days before Prince Andrew was born - the Queen declared that she had adopted the name Mountbatten-Windsor. She announced that it would be used by all her descendants who do not enjoy the title of His or Her Royal Highness.