Physical fitness can be measured by the volume of oxygen the body consumes while exercising at maximum capacity; also known as VO2max. You can measure VO2max by doing a bleep test or a submaximal test in the gym (on an exercise bike or treadmill). Theoretically, the more oxygen you use during high level exercise, the more energy you can produce. Those with higher VO2max values can therefore exercise more intensely and with more power.
Your VO2max is to a large extent determined by your genes (and the strength of your heart, lungs and muscles), but it can be increased by training. Numerous studies show that you can improve your VO2max by working out at an intensity that raises your heart rate for at least 20 minutes three to five times a week. It can also be improved by losing weight, assuming you lose fat whilst retaining muscle.
Nutritionally speaking, losing fat without losing muscle is all about eating sufficient protein every day. The recommended daily protein allowance is around 1 gram of protein per pound of body weight per day. If this need is not met through diet then your body may start to draw upon its own muscle tissue and your progress (and fitness) will suffer.