Over: Straight out of the bottle, into a glass with ice (as in "over ice"). Don't know if they're still teaching this at bartending school, most folks order "rocks" instead these days.
Shaken: Put in a shaker with ice and shaken until very cold. Chills better, but can result in cloudiness because shaking aerates the liquid.
Stirred: Put in a pitcher with ice and stirred until cold. Doesn't chill as quickly, may water down a bit more, but gives you a glass-clear result.
Mixed Drink: A drink that includes only one alcoholic ingredient and a mixer. If it's got two ingredients in its name joined with an "and" and comes in a glass with ice, it's a mixed drink.
Cocktail: A drink that (almost always) includes more than one alcoholic ingredient, and often more than two ingredients total (not counting garnish).
Up: Iced (shake or stir) and strained into a glass to remove the ice.
Rocks: Iced (shake or stir) and poured into a glass with the ice (in a really classy joint, strained into a glass containing new ice). Also commonly used in place of "over."
Single spirits can be ordered neat, up or over, tho it depends on the spirit if it's a good idea or not. Gin neat is usually a bad idea. Good whisky up or over will get you lynched in most places (tho Rye Over can be a delightful thing). Vodka (and flavored vodkas) work in any case.
Mixed drinks always come over ice, it's just assumed.
Cocktails can be ordered up or rocks, though some have traditional styles. That inverted-cone "martini" glass is an "up glass," and martinis are assumed to be served up. Daiquiris were originally up drinks, but evolved to rocks and then to blended. Same with Margaritas. A Mojito, on the other hand, always is a rocks drink (it's defined in the recipe).
If you don't put any vermouth in your martini, though, it's really just Gin Over or Gin Rocks, and it's time to admit you have a problem.