Aerobic activity is fueled by a process called oxidative phosphorylation, which processes fatty acids to provide energy. The body uses this as its long-term energy system; while at rest or doing low-effort work, this is the system that is used.
Anaerobic activity, literally meaning without oxygen, is the body's method of rapidly providing energy. Anaerobic metabolism can be divided into two main systems: ATP-CP (or phosphagen) and Glycolytic.
The ATP-CP system is the one used to provide very intense, very short bursts of energy. ATP supply is used up in 5-10 second, and creatine phophate stores can normally sustain effort up until around 30 seconds. I should note that the CP is actually rapidly converted to ATP, providing a "sparing" effect.
The Glycolytic system operates in slightly longer time intervals, from about 30 seconds up until about 90 seconds or so. This involves the breakdown of stored glycogen via the Krebs cycle in order to provide ATP. Lactate accumulation is a side-effect of this process; oxidative metabolism converts lactate back into a usable form (pyruvate) once activity ceases.
ATP-CP is most relevant to powerlifting, Olympic lifting, sprinting, and any similar high-effort, short-duration activity. Glycolysis is more important to moderate/high-effort, moderate-duration activity, such as martial arts, football, hockey, etc.
Though, that's just a generalization, since ALL sport activities will use all three methods to some degree.