Curiosity asks:
What is Prince Williams's and Prince Harry's last name?
I knew, somewhere in the back of my mind, that the royal family's last name is Windsor. However, this question prompted me to do a little bit of research.
I came across this lovely site that teaches you everything you ever wanted to know about royalty. According to their page on the British royal family, it turns out that the royals do not use last names in any official sense. I suppose they have no need for them.
The royal family is descended from the House of Saxe-Coburg-Gothe. However, during World War 1, people became concerned that this particular title sounded too German. The name Windsor was selected by King George V, and has been used ever since.
When Princess (now Queen) Elizabeth married Prince Phillip Mountbatten of Greece, she decreed that the last names would be hyphenated.
So, if you're addressing the Princes, HRH Prince William of Wales and HRH Prince Harry of Wales are correct. But technically, I suppose their last name is Mountbatten-Windsor.
CORRECTION: Upon further reading, I realize that anyone with the title Royal Highness has the last name Windsor, not Mountbatten-Windsor. Since William and Harry are both styled HRH, they would be Windsors. I do not know if this means Mountbatten-Windsor is reserved for those styled Majesty, or if it is for untitled descendants of the royal family. If you have further knowledge or information, please leave it in the comments, as I am now fascinated.