To supplement Peacedog's answerPeacedog's answer, the following screen shows up when you try to recruit new crew members, even if your roster is full:
In this case, you can see my existing eight crew members, and in the bottom right, the new Human, Maria, who would like to join my crew. Since I have already reached the maximum number of crew members, I can either dismiss
an existing crew member, or dismiss
the new recruit.
If you are having trouble deciding who to get rid of, you can hover over a crew member (and I'm kicking myself for not including it in the screenshot) to see their current skill levels. Keep in mind that new recruits in FTL:AE can come with pre-trained skills in later sectors.