Officially, their surname is Windsor. However, some descendants of Queen Elizabeth (including them) use the name Mountbatten-Windsor due to personal preference.

Their name is Windsor now but this was not always the last name used by the Royal family. Their name sounded too German so it was changed. By decree of the Queen their last name is actually Mountbatten-Windsor. “The royal family's official name, or lack thereof, became a problem during World War I, when people began to mutter that Saxe-Coburg-Gotha sounded far too German. King George V and his family needed a new, English-sounding name. After considering every possible name, from Plantagenet to Tudor-Stuart to simply England, the king and his advisors chose the name Windsor. To this day, the British royal family is known as the House of Windsor. When Princess Elizabeth (the current queen) served as a subaltern in the Auxiliary Territorial Service during World War II, she was called "Elizabeth Windsor." Elizabeth married Prince Philip of Greece, whose family name was Mountbatten, and eventually she decreed that most of her descendants would be called Mountbatten-Windsor. Princess Anne used this name in 1973 when she married Captain Mark Phillips. However, according to statements made by the queen, it appears that Windsor is still the official family name for any British royal who is styled "Royal Highness." The queen's youngest son, Prince Edward, the Earl of Wessex, has used the name Edward Windsor professionally. His wife calls herself Sophie Wessex.” http://www.royalty.nu/Europe/England/Windsor/