to me, a martini is a drink served in a martini glass that is a mixture of alcohol and vermouth.
first, a nitpick, you are never supposed to shake alcohol, it should always be stirred. shaking "bruises" the alcohol. I dont make this stuff up, i just learn from the snobby alcoholics/bartenders.
second, against what a small portion of people here are saying, i enjoy vodka to gin. i dont think using vodka disqaulifies the drink as a "martini". someone further down posted about expensive vodkas. ill elaborate. essentially, vodka is pure grain alcohol, from potato or certain wheats. it just depends on the maker, but the alcohol is clear, unlike whiskey, the american grain alcohol. the vodka is distilled, the more times the better quality and the smoother the taste i find, to almost pure. pure is 200 proof, or 100% alcohol. then the alcohol is "cut" with water. again, the better the water, the smoother the taste. i find smirnoff to be obnoxious, absolut to have a bite and off smell, stoli (stolichnaya) and skyy to be excellent for the price ($12-15). grey goose is like a more expensive, slightly smoother version of those, belvedere, made popular by hip hop stars is excellent, but my personal favorite is belvedere's cousin, Chopin. thats my drink of choice ($30-35). of course, go to a club and expect to pay $100 and up for any of those three.
gin is made from the berries of the juniper bush, which i find ironic since most juniper bushes smell like cat piss. thank louis (i dont remember which number, google for it), the king of france for coming up with that one. he wanted it to be like a wine, and thats what he got, an extremele fragrant drink. i find most are similar, but the more you spend, the smoother the drink. tangueray 10, beefeater, bombay sapphire, all excellent.
chilling glasses, and the alcohol is always a good idea. the authors method of .25 to .50 ounce vermouth swirled to coat the glass and discarded is the "upscale" technique. the easy way is just to put .25 to .50 ounces and then the alcohol on top.
the martini defaults to gin unless "called" vodka by saying vodka martini. the classic is gin and dry vermouth, there is the sweet martini, in which you substitute sweet vermouth for dry vermouth but thats not done very often and i dont recommend it, and served with olives. the gibson is gin (can be called vodka) and dry vermouth, and served with pearl onions. the gimlet, vodka and lime juice (call fresh lime juice) and dry vermouth. "dirty" martini, gin (call vodka), olive juice, dry vermouth. the manhattan, the only martini not made with gin or vodka. the manhattan is whiskey or bourbon and sweet vermouth. you can call a dry manhattan in which you substitute dry vermouth, or "the perfect manhattan" in which you use 1/2 sweet and 1/2 dry vermouth, and whiskey or bourbon. of course theres the cosmopolitan, vodka, lime juice, triple sec, and cranberry juice. some really good variations come from using chambord. and the ever popular (thanks hip hop culture) appletini, vodka and apple pucker.
some places to learn more - http://cocktails.about.com/library/recipes/blmartinimenu.htm
and my personal favorite - http://www.webtender.com