Condottieri is a fancy Italian word that means "mercenaries," or soldiers who are paid to fight for someone else. Think of them like some big kids who are really good at playing war games and get paid for it.
In the olden days of Italy, there were so many little kingdoms and city-states where people were fighting all the time. These different states all needed armies and soldiers to fight for them, and that's where the condottieri come in. They were like the professional soccer players of warfare.
The Italian princes and nobles would offer lots of money to these mercenaries to fight for them, because it was often easier and cheaper than keeping a standing army themselves. The condottieri would then hire other soldiers, known as "condotta," to form their own little armies.
So, in short, condottieri were professional soldiers who were hired to fight for Italian states, kingdoms, and princes. They were like the ultimate "hired guns."