What do we call her, the newly married wife of Prince William?
The official title of the former Catherine Middleton is Her Royal Highness The Duchess of Cambridge, according to the statement issued by Queen Elizabeth's press secretary today. The Queen made William "Duke of Cambridge, Earl of Strathearn and Baron Carrickfergus," combining English, Scottish and Northern Irish titles. With those titles, William is more formally known, according to protocol, as His Royal Highness Prince William, Duke of Cambridge -- the dukedom being the highest rank of the three new titles. So, on marriage, Catherine Middleton became the Duchess of Cambridge.
Calling a woman "Princess So-and-So" is reserved for royal-born women.
But what about Princess Diana? The quick answer is that while many referred to the wife of Charles, Prince of Wales, as Princess Diana (or even Princess Di), that title was not a correct usage according to protocol. She was Her Royal Highness the Princess of Wales when married to Charles, Prince of Wales. After the divorce, the HRH or Her Royal Highness was taken away, but she was permitted to style herself "Diana, princess of Wales."
On this site, I use the common usage "Princess Diana" in many articles, because so many people who are looking for the information look for it under that title. (Similarly, I've used "women's suffrage" instead of "woman suffrage" even when it's anachronistic, because most people research looking for that rather than "woman suffrage.")
Today, Camilla, second wife of Prince Charles, is technically Princess of Wales, but she prefers to use the title Duchess of Cornwall, based on one of Charles' many other titles, the Duke of Cornwall. Whether she will have the title of Queen if or when Charles succeeds his mother and becomes King is another matter entirely, a matter of some conflict -- but both Charles and Camilla are said to be leaning towards a "yes."
Meanwhile, just as Diana was generally called Princess Diana or Princess Di, it's likely inevitable that Catherine will be called Princess Catherine (I heard an interview this morning with one of the royals calling her just that) or even Princess Kate.
Oh, and one more thought. Though Catherine might one day become Queen Catherine, if or when William becomes King -- but many Queens Consort are known to history by their names before marriage. Thus, we remember Elizabeth Bowes-Lyon, Anne Hyde, Catherine of Valois, Eleanor of Aquitaine, and so on. So Catherine may be known to history simply as Catherine Middleton.