Primogen is a term used to describe Camarilla officials that, at least in theory, serve as the representatives of their respective clans to the prince of a city ruled by the sect. The term is both singular and plural, and refers to both individual representatives and to the Primogen council as a whole.
As with many Camarilla offices, the Primogeniture predates the very existence of the sect, and was pretty common during the Dark Ages. Theoretically, a primogen is the oldest vampire of his clan, and serves as a representative and voice for the clan in city affairs. In general, the ruling vampires of the city value the Primogens and their opinions. They are called in to consult on decisions, and their recommendations carry great weight.
In the modern era, the officeholder is usually decided within the clan proper by voting, violence, or someone higher up giving the orders, rather than simply comparing ages. Primogen traditionally make up a council that is the second most powerful institution in the city after the Princedom.
The power of the primogen varies with the power of the vampires that make it up: in some cities they are a rubber stamp for the prince, while in other cities they manage all affairs.
Sometimes certain Clans get multiple primogen, sometimes they get none. Sometimes a prince counts as the primogen of his clan, sometimes he doesn't. In most cases it is going to depend on who in the city both wants and is capable of demanding a seat. If there's two Ventrue elders in town besides the Prince and they both want a primogen seat, it'll be hard not to grant it to both.