When you nab someone, you catch the person in the act of doing something wrong. A police officer, for example, might say that she loves to nab shoplifters.
You nab when you apprehend a bad guy, or when you suddenly grab something: "Hey, you can't just nab a slice of pizza off my plate!" It's most common to use nab when a criminal is caught, as when the FBI nabs a suspected terrorist or an officer nabs a would-be graffiti artist. Nab probably came from an earlier word, nap, "to seize or catch," which is now obsolete except in the word kidnap.