Fat molecules consist of three fatty acid chains connected by a glycerol backbone. Fatty acids are basically long chains of carbon and hydrogen and are the major source of energy during normal activities.
Fatty acids are broken down by progressively cleaving two carbon bits and converting these to acetyl coenzyme A. The acetyl CoA is the oxidized by the same citric acid cycle involved in the metabolism of glucose. For every two carbons in a fatty acid, oxidation yields 5 ATPs generating the acetyl CoA and 12 more ATPs oxidizing the coenzyme. This makes fat a terrific molecule in which to store energy, as the body well knows (much to our dismay).
The only biological drawback to this, and other, forms of oxidative metabolism is its dependence on oxygen. Thus, if energy is required more rapidly than oxygen can be delivered, muscles switch to the less efficient anaerobic pathways. Interestingly, this implies that an anaerobic workout will not "burn" any fat, but will preferentially deplete the body of glucose. Of course, your body can't survive very long on just anaerobic metabolism...it just can't generate enough energy.