Brightness is the taste of sharp notes in the coffee, not unlike the tannins in wine. These flavors are sensed in the front of the mouth and are caused by certain kinds of acids. (The term acidity is also used frequently to describe these flavors, but the word carries a lot of misplaced negative connotation; it's unrelated to bitterness or anything that would make you grab a roll of antacids.)
Brightness is a desirable quality that gives a coffee its intensity or bite. It's particularly critical for milk-based espresso drinks, such as lattes and cappuccinos, as the flavor of the coffee might not otherwise hold up to dilution among the smooth sweetness of steamed milk. Too much of the wrong kind of brightness, however, and an espresso can taste sour.