No, its another person just like you (or a fund, but I digress). It is rarely the company itself buying the stock (though buybacks do happen, but I digress again).

The number of buyers is always equal to the number of sellers - that is what the stock price actually is (the price at which buyers = sellers). The price is constantly adjusting to keep that statement true.

So, for example, if you want to sell stock at $100/share, maybe no one wants to buy it for that - they only want to buy at $80/share. Someone else, who really wants to sell will offer $90/share and someone else who really wants to buy will do that. The stock price is now $90. You still have your share at $100 and the guy wanting to pay $80 still doesn't have his, but since the number of people willing to sell at $90 = the number of people willing to buy at $90, that is where the price evens out.