In May, Prince Harry and Meghan Markle became parents for the first time. They brought in little Archie, who is a copy of her father. But let's make you something clear: Archie is not a prince. He did not receive this title as his cousins, as in 1917 King George V decided that the royal title would only be retained by the rulers (Prince Charles, Prince Edward and Princess Anne), the children of Charles's greatest ruler Prince William and Prince Harry in this case), as well as the great baby of Prince William's greatest baby. Which means that Prince Harry's children do not take the royal title.
You rightly ask why Princess Charlotte and Prince Louis took the title, though not William's long-lasting children? In 2013 Queen Elisabeth (who has all the power to change the rules of the kingdom) decided that the title would be taken by all William's children.
Meghan and Harry might have asked the queen to do this, but they did not do so because they wanted Archie to live a normal life. But what is Archie's title? He is Master, Master Archie! The Master is a title used for a royal descent who is too young to be called the Master (gentleman).
We recall that Archie is the seventh in the throne list and the ranking for the throne is such:
1. Prince Charles
2. Prince William
3. Prince George
4. Princess Charlotte
5. Prince Louis
6. Prince Harry