The company itself issues you the options, usually as a form of compensation because you're an employee, etc.
That option is a certificate that gives you the ability to purchase a share of the company at a specific price (the strike price). Usually when people say "buy their options" or "exercise their options", they are referring to buying the _stock_ that their options gave them the ability to purchase.
So if I join a startup, they grant me 100 options with a strike price of $0.50, and I decide to exercise them, I would write the company a check for $50 and get 100 shares of the company in exchange.
