Some info from a former bartender:
All chilled, straight spirit cocktails--that is cocktails made entirely of booze with no non-alcoholic mixers, juices, etc.--should always be stirred. The primary reasons for this are so that the bartender has total control over the temperature of the cocktail and the amount of dilution being introduced. The most commonly-ordered straight spirit cocktails are the martini, the Manhattan, and occasionally the Negroni.
The proper method for making a straight spirit cocktail is to first add the ingredients to a mixing glass. The base spirit should be added first, then each other ingredient in descending order of volume. (Using a Manhattan as an example, you would measure out and add the whiskey, then the sweet vermouth, then the dashes of bitters.) Gently add enough ice to fully cover the the spirits, and use a barspoon to stir until desired temperature and dilution is achieved. Cover the mixing glass with a Hawthorne strainer, and strain the cocktail into a chilled glass. Some bartenders like to double-strain (a Hawthorne strainer held to the mixing glass, then pouring the liquid through a mesh strainer into the cocktail glass) just to make sure no small chips of ice make it into the finished drink.
What does this have to do with Bond and "shaken, not stirred"? Well, Ian Fleming knew how a martini was made. It was, without question, stirred. He wanted something to signify that Bond operated a little outside the normal rules, that he was slightly rebellious. Having Bond order his martini shaken was significant because it was a break from tradition and expectation, a break from the "correct" way the drink was prepared.
Then the line made it into the films, and over fifty years the general public fell into assuming "shaken, not stirred" is the way they should order their martinis. But it's not. In fact, unless you specify "shaken," a high-end bartender will never prepare the drink for you that way. They will always stir it.
Side note: People have also become so accustomed to the idea of a "dry" martini that many high-end bartenders interpret all vodka martini orders as "dry" even if not specified. When you add vermouth--in any amount--many customers will tell you it tastes wrong and ask you to re-make the drink. When I was bartending, an order for a vodka martini meant three ounces of vodka, stirred, and strained into a martini glass. No vermouth at all.
It's interesting to see how time and culture have affected the way this classic cocktail has evolved.