See also: What's My PHX Coffee Shop? A Flowchart.
The Basics: Brewed Coffee is the black or brown liquid that many people have in mind when they think of coffee. If you want the basic deal, this is what you should ask for.
Brewed coffee can be prepared a billion ways, and many shops offer an array of different brewing methods. Pourover brewing is just what it sounds like: hot water is steadily poured over a "bed" of coffee grounds, some beautiful chemical mumbojumbo happens, and you wind up with a cup of coffee. Pourover brewers generally utilize some type of filter (usually paper). The filter traps a lot of little coffee particles, which means the resultant cup is generally clean and articulate.
If you want a heftier, fuller-bodied cup of coffee, ask if your favorite shop offers any immersion brew methods. The most mainstream of the immersion brewers is the French press, but an immersion brew is any method of brewing coffee wherein coffee grounds are steeped in water for a period of time (generally four to five minutes). The grounds are then filtered out, generally by a mesh screen of some kind. This screen traps the bigger particles but lets more of the little ones through than, say, a paper filter would. Immersion brewed cups have big, fat bodies and usually have a somewhat viscous, oily mouthfeel.
Espresso is a method of brewing coffee by the application of pressure. Espresso is a tiny little drink (less than 2 ounces) with a huge set of jargon-y terms to be conquered. It can be excellent on its own, but espresso also serves as the base of a number of other espresso drinks.
A lot of people panic when they look at a cafe menu and gravitate toward an Americano without really knowing what it is. This drink is the whiskey soda of the coffee world: espresso, plus water, equals an Americano. If you're new to coffee and want to ease into exploring espresso, Americanos are a great place to start. You'll taste all the flavors of the coffee, but spread out a little bit so they're easier to interpret.
I like to break down coffee flavor traits into three basic categories for new drinkers. By this, I don't mean flavor additives, but rather inherent qualities that a coffee naturally possesses. These are in no way comprehensive, but they can help ease communication between barista and customer.
As a general rule, try to stay away from using terms like "bold" or "robust." These terms are vague, and don't give a barista much information about your preferences. Instead, talk about body: Do you want something full-bodied or a little less so?
If you like a tangy, fun coffee, ask your barista for something bright, fruity, or juicy. "Bright" is how coffee folk describe cups that are pleasantly acidic, but not overwhelmingly so.
If you want a coffee with candy-bar like qualities (think milk chocolate, peanut butter, caramel), ask for a sweet, balanced, full-bodied cup.
If you want a hefty coffee with a big, fat body, you have a few options. Ask for something dark chocolatey, earthy, or natural-processed. We'll talk about processing a different day, but naturally-processed coffees have a tendency towards rich, concentrated, red wine-type flavors.