What you're seeing is the great people available right now.
There are 3 great people of any type for each era (some GPs can't be earned normally until later eras so that's why engineers or musicians start later era than ancient.)
There's as many prophets as the max number of religions available (based on map size. Std size is like 5? 6?)
When someone gets enough GP points, they unlock the person shown. When a great person is recruited, a new one takes their place. So it's sort of a "race" for the ones you want. Each has a unique effect. Some scientists, for example, might grant eurekas (tech boosts) while some may grant a bonus to university buildings or perhaps science based on where they are activated (science per mountain they are standing next to, for instance.)
You can also skip a great person, but be warned; it can take the AI a long long time to earn a specific GP category, so make sure they are actually making progress in whatever area your skip. Further they can be bought with gold or faith and that cost is based on how close you are to getting them. Note that the Ai will sometimes buy out a great person if it wants it and you are both close to getting it!!