It gets kind of complicated and its use is more often unofficial than official.

When Queen Victoria married Prince Albert her children were deemed part of his and not her house as she, being a female Hanoverian, could could not pass the name of Hanover on to her children and her Son, Edward VII became King as a Monarch of the House of Saxe-Coburg & Gotha.

This was changed to Windsor by his son George V due to the anti-German feelings created by the war with Germany in 1917.

Mountbatten had been similarly changed from Battenberg for the same reason and Prince Phillip had adopted the Mountbatten surname as his mother has been one - it was somewhat easier to use than Schleswig-Holstein-Sonderburg-Glücksburg - the House of his father!

However, when she ascended the throne in 1952, the Queen announced that both she and her descendants would still be known as members of the House of Windsor. Prince Phillip was reportedly far from happy about this at the time.

In 1960 the Queen decreed that certain of her descendants, specifically those who did not hold royal styles and titles could use the Mountbatten-Windsor surname. This has not prevented, however, some of those who have such styles and titles from using it in some circumstances. Princess Anne signed herself "Mountbatten-Windsor" in the marriage register for example.

All grandchildren of the Monarch through the male line are Princes and Princesses automatically (thus, strictly speaking, they do not qualify for using Mountbatten-Windsor) and take their titles from their father. Hence Beatrice is "of York" for example because of her father's title.

Male great grandchildren through the male line do not qualify for royal rank and title and could therefore use the name. Currently there are none so descended from the Queen.

It's less fair on the female descendants, however, as Princesses are limited to daughters of the Monarch and granddaughters of the Monarch through the male line. When they marry, whilst they retain royal rank themselves, they cannot pass that on to their children who can only take the surname and rank of their father. Hence the children of Princess Margaret are not royal (although still counted as part of the royal family and in the line of succession) although they do have non-royal titles due to their father's rank as he was created Earl of Snowdon, But the children of the Princess Royal have no title at all as Mark Phillips took no title (but they still are in the line of succession). So they are simply Peter and Zara Phillips.

Any clearer or clear as mud?