Buying & Selling Stocks - Anatomy of a Trade

How does the buying and selling of stocks (a stock trade) work?  There are three parties involved in a trade:

The Seller: This is the person or company who owns the stock but wishes to sell it. 

The Buyer:  This is the person or company who wishes to puchase the stock. 

The Broker:  These are the people who match buyers and sellers.  While the buyers and sellers will never meet, or even know each others' names, the broker earns his living getting buyers and sellers together. 

Let's walk through the anatomy of a trade, one step at a time..