Most cats are "Warriors". (You can also be, Medicine cat, kit, apprentice, queen, elder, deputy, leader. More on that later.)
When you are a warrior, you hunt for the clan, fight to protect it, train younger cats to become warriors. When you are a warrior your name will be two parts. Here are some examples: Stonefur, Fireheart, Silverstream, Lionblaze etc etc. Your leader will give you a ceremony, changing you from apprentice to warrior.