Anaerobic literally means "without oxygen." This doesn't mean the sensation of gasping for breath after you've climbed 15 flights of stairs; it is a little more complicated than that. It really just boils down to what fuel your muscles use to do whatever it is you are asking them to do.
For contrast, let me explain aerobic training. When you go for a long run, oxygen is released to metabolize, or make, energy for the muscles. So aerobic means with oxygen and is generally any activity lasting more than two minutes. When you exercise anaerobically, like sprinting, you're burning energy too quickly to use oxygen as fuel and your muscles turn their own stored glycogen into energy, without the help of oxygen. There is a price for this convenience; you only have 90 seconds to two minutes worth of stored energy in your muscles. Plus, as by product in that energy conversion your body makes lactic acid, which is responsible for that burning sensation you get in your muscles when working at high intensity.
Anaerobic conditioning helps with speed, strength and agility. Three very important fitness elements for athletes. In fact, anaerobic training was considered essential for elite athletes, but not important to average fitness folks. Times sure have changed because interval training, a mixture of aerobic and anaerobic exercise, has made its way to the masses. To add anaerobic training, adding sprinting intervals in your runs, bike rides or swims, you do need to have an aerobic fitness foundation.
Strength training, from push ups to leg presses, is also considered anaerobic.