How does a mutual fund buy shares in a company?

Do they call their broker and say "put in an order for 100,000 shares at $23.51?" Or do they have a broker in house thats connected to NYSE floor or that belongs to the electronic market. For that matter what does my broker do after I call him. I mean who does he call? And then who does the guy my broker called call. For that matter where are my shares, I never see them. Am I actually the shareholder or is my broker the shareholder?