"Aerobic" in physiological terms means "in the presence of oxygen", and in the context of exercise it means those modes or intensity levels that utilize oxygen as a prime source of fuel for the working muscles (oxygen that is transported by the blood).
"Anaerobic" in physiological terms means "not in the presence of oxygen", and in the context of exercise it means those modes or intensity levels that rely exclusively on limited, internal energy stores to fuel the exertion.
Aerobic exercise tends to be at a lower intensity level than anaerobic exercise, and thus can be sustained for much longer periods because oxygen is continually being used to combine with other energy substrates in the body to fuel the movement. When one raises the intensity level to an all-out effort, one starts to tap into the other energy stores; these all-out efforts cannot be sustained for more than a few seconds because these particular energy systems are depleted very quickly. Think of lifting the absolute heaviest weight you can, ONCE, or think of sprinting as fast and hard as you absolutely can.
Cathe's Interval Max I is a good example of a workout that brings you close to your anaerobic threshold, especially the meanie intervals, #2, #4, #5, #8 and #10.
Hope this helps.
A-jock