A martini is a simple cocktail of gin and vermouth -- with a proportion of one part dry vermouth to four parts gin; it's usually garnished with a green olive or a lemon peel. The cocktail's origin is a bit of a mystery. Some sources credit the introduction of the martini to the town of Martinez, CA, others to Randolph Martine, a New York Judiciary at the turn of the 20th century. Whatever its origins, prohibition is often attributed for this drink's rise popularity due to how easy it was to illegally manufacture gin. With time, the martini's popularity waned in the face of newer cocktails and wine spritzers, but the martini experienced a resurgence in the 1990s and it looks like the cocktail is here to stay -- and it's not hard to understand why.
If you order a dry martini it means that you'd like less vermouth in your cocktail. A very dry martini would contain little to no vermouth; one might just roll the vermouth in the glass so that a thin film coats it.
As you might imagine, ordering a wet martini will get you more vermouth in your cocktail than the standard five-to-one ratio.
A shaken martini means that the drink is mixed in a cocktail shaker with ice. This is now the more popular way to order a martini, as most cocktails are made with a shaker.
Stirred martini is a more traditional way of making martini drinks. Some experts argue that the only way to drink a martini is stirred instead of shaken, so as not to dilute the flavor, but it really comes down to preference.