A "dry martini" uses less dry vermouth than normal, perhaps a dash or lace of the glass. Similarly, a "wet martini" refers to a martini that uses a greater amount of Vermouth. A dry martini originally meant one made with white vermouth rather than sweet and had nothing to do with the amount of vermouth per volume of gin. Before WWII the ratio was usually 3 or 4 to 1 gin to vermouth.
A "50-50 martini" uses equal parts gin and dry vermouth.
A "perfect Martini" is technically one made with a mixture of equal parts dry and sweet vermouth, although in many bars the term is misused as a qualitative one.
An "upside down martini" or "reverse martini" uses more vermouth than gin.
Variations on the gin or vermouth
A Vesper is a variation also favoured by James Bond, which is made with three measures of gin, one measure of vodka, and half a measure of Kina Lillet aperitif, shaken until ice-cold, and with a large, thin slice of lemon peel for garnish.
A Vodka Martini substitutes vodka for gin, and often uses lemon rind as the garnish. This is the most common variation. It was made famous by the James Bond movies as James Bond's favourite beverage. He is known for requesting it "shaken, not stirred"
Chilling and dilution
A martini should be well chilled. Conventionally they are carefully stirred with ice, then strained, to leave as little ice as possible in the glass afterwards. A martini that is 'shaken, not stirred' would be chilled, but it would also tend to break up the ice, leaving the smaller fragments to melt and so dilute the drink. Bond orders his martinis diluted.
Sometimes the term "martini" is used to refer to other mostly-hard-liquor cocktails such as Manhattan, Cosmopolitan, and ad hoc or local concoctions whose only commonality with the drink is the cocktail glass in which they are served. Chefs with a more whimsical bent are even producing dessert "martinis" which are not a drink at all, but are merely served in martini glasses. Another popular form is the espresso martini, popular in restaurants as a dessert. Many variations exist but most involve shaking an espresso shot with the ingredients and served in a chilled martini glass. By shaking a fresh espresso shot it creates a hard layer of crema which is garnished with three coffee beans in the centre. A "Porn Star Martini" is a popular variation of a vodka martini. Traditionally, the vodka is vanillaflavoured, and is mixed with Passoã and passion-fruit juice before being served in a martini glass, accompanied by a shot glass filled with prosecco.