Everyone's metabolic rate is slightly different, but on average, it takes about 13 calories per pound of body weight per day to maintain that body weight for a person with minimal physical activity. Physical activity increases the metabolic rate, so in addition to the actual calories burned during the activity, the overall metabolic rate increases to the point that instead of 13 calories per pound per day, the body might need 15 calories per pound per day.
To make things more confusing, starvation will slow the metabolic rate. Thus, if one goes on a crash diet and restricts caloric intake too much, the metabolic rate will slow to the point that weight would be lost more slowly than it would have been if more calories were consumed.
A pound of fat provides 4081 calories (9 calories per gram x 453.5 grams per pound). Thus, to lose a pound of fat, you must expend about 4081 more calories than you consume. This pound of fat, though, will be more than a pound of body weight, since there are other things in the body, primarily water, in equilibrium with that fat that would also be lost when the fat is burned.
How long it takes for the weight loss to show up depends on a lot of factors. Under heavy exercise, the body cannot utilize fat as an energy source rapidly enough to keep up with the demand. Initially, you will probably lose more than a pound, since the primary energy source is glycogen, and glycogen only provides 4 calories per pound. If all of the glycogen is used up and you are still exercising at a high intensity level, you will lose some muscle before you totally bonk. Muscle also provides about 4 calories per gram and is not where the body wants to get its energy; however since fat cannot be utilized quickly enough, if the glycogen is depleted muscle will be wasted.
Also, exercise causes water loss, although this will be quickly replaced once the water intake catches up with the water lost. Eventually, fat will be utilized to replace the glycogen, and a new equilibrium will be reached. By this point, the 1000 calorie deficit will have resulted in about a one-quarter pound weight loss, asuming that the food intake did not increase.
The bottom line is that if calories expended are more than calories consumed, weight loss will happen. Without measuring expired CO2 to get a good idea of the metabolic rate, it is difficult to determine how many calories are being expended.
