In 1917, King George V made a rule that great-grandchildren of the monarch would not get to be princes or princesses, apart from the eldest son of the eldest son of the Prince of Wales.
The current Prince of Wales is Prince Charles, meaning Prince William's eldest, George, is a prince.
William and Kate's other children, Charlotte and Louis, are princess and prince respectively after orders issued by the Queen back in 2012 which extended the title and style to all children of the Prince of Wales' eldest son.
This will all change when Prince Charles becomes King though.
Archie and Lili will then go from being the monarch's great grandchildren to the monarch's grandchildren, and will then be considered a prince and princess respectively.