Actually, it's important to do both. Of course, do not eat a large meal that will make you uncomfortable during your workout, but you MUST have SOMETHING before working out. The number one reason I see people pass out at the gym I work in is because they did not have breakfast before they came in.
Your body needs fuel to propel the activity you are doing. Fat is a slow-burning fuel source. Trying to exercise on fat stores alone will result in a less effective, less efficient workout. Working out lowers blood sugar. If your blood sugar is already low because you have not eaten, exercising could bring it to a dangerously low level resulting in weakness, dizziness, loss of endurance, and potentially, black outs. So, eating before the workout is important.
Eating after the workout is equally important for recovery. Eating a 4:1 ratio of carbs:protein within 30-45 minutes after a workout speeds up recovery time, decreases soreness and helps your body perform better at the next workout.
J Appl Physiol. 1992 May;72(5):1854-9.
Proper nutrition is key in optimizing your workouts. For any other nutrition questions, feel free to send me an email directly- jessi.cano@live.com or post another question.