It is another person/organization. It could be the company itself but the company is rarely a big purchaser of their own stock on a day to day basis. The entire purpose of the stock MARKET is to provide a place for buyers and sellers to meet, agree on price and transact their goods (stocks in this case). In the past this was an actual physical space (like a real market) and brokers would shout out buy and sell requests. Today it is mostly electronic trading on computers.

A modern stock market is designed to be really fast and efficient in transaction. The major stock markets has some things working in the background. For many stock markets, some "special" companies will "make the market" for some particular stock. They will be the counterparty (ie if you buy they will sell and if you sell they will buy) to transactions for the company they are the market makers for. Since buys and sells generally balance out over time, these companies just make transactions work quicker.

For small retail investors dealing with a broker and small transactions, the brokerage themselves can act as a counterparty (essentially you're selling and buying from your broker and the broker consolidates all these customer transactions and balances it out by the end of the day by going to the market and squaring it off)