First, a quick definition from the Specialty Coffee Association of America: 

Espresso is a 45ml (1.5 ounce) beverage that is prepared from 7-9 grams of coffee through which clean water of 192° - 198° F (88° - 92° C) has been forced at 9-10 atmospheres of pressure, where the grind of the coffee has made the brewing "flow"* time approximately 22-28 seconds. While brewing, the flow of Espresso will appear to have the viscosity of warm honey and the resulting beverage will exhibit a thick dark gold cream foam ("crema") topping. Espresso is usually prepared specifically for, and immediately served to its intended consumer. 

*Brewing "flow" time is counted from the point in time that the espresso begins flowing from the bottom of the portafilter spouts. Many espresso machines have a distinct "pre-infusion" cycle where a small quantity of water is pumped into the grounds and allowed to sit for several seconds before the pump is reactivated to finish the brew. The prescribed method of "timing" shots when the flow begins to appear at the portafilter spout can be universally applied for all machines, no matter the pre-infusion cycle.

The rest of the specialty coffee drinks are mostly based on espresso.  For example, a cappuccino is an espresso with milk and foam.  A latte is an espresso with just steamed milk and very little, if any, foam.  Pressure-brewed coffee is made like an espresso but with more water.

A common misconception is that espresso is a certain type or roast of bean.  In fact, you can use any bean or roast that you like.  The definition refers to the brewing methods rather than the coffee itself (fast, served immediately and under pressure).  The only right choice of bean is the one that tastes best to you!

So now that you know what an espresso is, the next step is to learn how to go about making it.  There are several different styles of espresso makers, with a wide range of prices and features.  Here is a brief introduction.