The term is made famous by Ian Flemmings character James Bond. In his original novel; Casino Royale.

"Wikipedia" wrote:

In Fleming's novel Casino Royale, it is stated that Bond "watched as the deep glass became frosted with the pale golden drink, slightly aerated by the bruising of the shaker," suggesting that Bond was requesting it shaken because of the vodka it contained. Prior to the 1960s, vodka was, for the most part, refined from potatoes (usually cheaper brands). This element made the vodka oily. To disperse the oil, Bond ordered his martinis shaken; thus, in the same scene where he orders the martini, he tells the barman about how vodka made from grain rather than potatoes makes his drink even better.

Scientists, specifically biochemists, and martini connoisseurs have investigated the difference between a martini shaken and a martini stirred. The Department of Biochemistry at the University of Western Ontario in Canada conducted a study to determine if the preparation of a martini has an influence on their antioxidant capacity; the study found that the shaken gin martinis were able to break down hydrogen peroxide and leave only 0.072% of the peroxide behind, versus the stirred gin martini, which left behind 0.157% of the peroxide.[10] Thus a shaken martini has more antioxidants than a stirred one. The study was done at the time because moderate consumption of alcohol appears to reduce the risk of cataracts, cardiovascular disease, and stroke.

...many bartenders insist that any cocktail that involves nothing but transparent ingredients - such as martinis, manhattans, and negronis - must be stirred in order to maintain clarity and texture. The former is an aesthetic concern, the latter a matter of culinary taste. Shaking a drink is quite violent, and necessarily introduces air bubbles into the mix. This results in a cloudy appearance and a somewhat different texture on the tongue when compared to a stirred drink.

Hope that helped.