an official position as a member of a committee, council, board of directors or legislative body etc.
He gave up his seat on the Legislative Council.
having a seat on a stock exchange, especially in the US, means that you are a member of that stock exchange and are allowed to buy and sell shares
These seats are a sought-after commodity as they confer the ability to directly trade stock on the NYSE (=New York Stock Exchange).
the site or location of something, especially something official or a business entity etc
The voyage was booked through Sniegon Travel, a company whose seat is located in Poland.