Great People can be earned by generating enough of the right type of points. They are visible to all players, and they are visibly competed for between all players. Only major civilizations can earn Great People, not Barbarians or city-states. Each type of Great Person has its own points which are tracked separately. Great Scientist Points cannot count as Great Artist Points, and so on. The number of points needed to earn a particular Great Person depends on that Great Person’s era. A Renaissance-era Great Scientist will cost much less Great Scientist Points than a Modern-era one. The points come from a variety of sources.
Each district, and the buildings within it, provides points towards its associated Great Person type. This is the slow but steady source of points. Districts also allow a city to use Production towards projects which, when completed, contribute Great Person points.
Players can also spend Gold or Faith to patronize a Great Person, earning that person instantly. The patronage cost is based on how many more Great Person points the player would need to recruit that person normally.