The Basics
Each of the nine Great Person classes accumulates its own point total empire-wide (not per city). Once you have enough points, you can recruit the next available individual in that class — each one belongs to a specific era, and the cost to recruit rises as the game progresses through eras, even if earlier-era individuals are still unclaimed. Every distinct Great Person can only be recruited once per game, across every civilization at the table, so rival civilizations are genuinely competing with you for specific historical figures, not just a generic point total.
Beyond generating points naturally through buildings, you can also recruit a Great Person outright through Patronage — spending a lump sum of Gold or Faith (Faith is typically the cheaper option) to acquire whoever is next in that class's queue immediately.