Gin
Gin was created by a Dutch chemist, Dr. Sylvius in mid 1600s. His intention was to invent a medicine that would clean blood for kidney disorders. He called it "genever," meaning juniper in French. Gin is a neutral grain spirit infused with various botanical "flavorings". Most Gin is colorless, however, some brands may be golden due to their aging process in wooden barrels. Most Gin however, is not aged. There are different kinds of Gin; dry Gin, London dry, Holland, and many flavored types. Dry Gin being the most popular, and is the most called for when making Martinis.

Vermouth
There are two types of Vermouth. French and Italian. Italian is also known as "Sweet" Vermouth, and French is generally known as "dry" Vermouth. Vermouth is made by infusing a base wine with various botanicals. When it comes to Martinis you always assume Dry Vermouth is to be used. Some drinkers will try to tell you to omit the Vermouth, or some humorous tip like "just whisper Vermouth as you stir" or "just wave the Vermouth bottle over the shaker". While this is a fine way to drink Gin, without Vermouth, it is NOT a Martini.

Green olives
Green Olives are regular Olives which are picked before they are ripe, and then cured in brine. They come in many sizes and "flavors", from garlic to anchovy, but the standard for Martinis is a pimento stuffed olive

Bitters
The recipes for bitters are closely guarded by the makers. There are many brands, and types, but the "industry standard" is Angostura. Bitters are mildly alcoholic, very strong flavored, And most often slightly orange flavored due to most recipes containing Orange Oil. Usually only used in one or two drops, a little goes a long way.

Terms

Shaken
If you order your Martini "Shaken" it means mixed by shaking in a Cocktail shaker usually half filled with ice. Shaking gives you a colder drink with a more mild flavor. James Bond likes them this way, the Thin Man likes them this way. I like them this way. 'nuff said.

Stirred
If you order your Martini "Stirred" it means mixed by stirring in a Cocktail shaker usually half filled with ice cubes. Stirring gives a slightly stronger Gin flavor to the Martini.

Dry
If you order your Martini "Dry" it will be made with less Vermouth.

Over
If you order your Martini "Over" it means strained into a glass over Ice.

Dirty
If you order your Martini "Dirty" it will contain a splash of brine from the Olives.

Vodka
If you order a Martini, it should be made with Gin. If it is not, go to another bar, with a real bartender. But if you order a "Vodka Martini" it will be made with Vodka instead of Gin. Always remember to specify

Gibson
A Gibson is a regular Martini, but garnished with a pickled Onion instead of an olive.