Owning a seat on the New York Stock Exchange (NYSE) enables a person to trade on the floor of this stock market, either as an agent for someone else (floor broker) or for one's own personal account (floor trader). Prices for seats on the exchange are determined by supply and demand, and they have ranged from $4,000 to over $2,500,000.
However, to own a seat, a trader or broker must meet more criteria than simply being able to afford it. Prospective owners of the seats must go through a stringent review process and, once accepted, they are required to maintain high levels of compliance and ethics as the NYSE and governmental regulators constantly review members.
The phrase "owning a seat on the exchange" actually has very literal origins. Up until 1871, trading on the exchange was done in a "call market" fashion, which is a system by which only one company's stock trades across the whole exchange at any one time. Trading members would sit in the assigned seats they owned and participate in the buying and selling of desired stocks as they were called for trading. After 1871, the trading of stocks became simultaneous and the floor trading that we're accustomed to today became the norm.
For more about trading, read "Getting to Know Stock Exchanges" and check out our Brokers and Online Trading tutorial.