Of course we need both fat and calories. Children especially need fat during development. The brain consists of a high percentage of fat and will not develop properly without a proper amount of fat usually obtained from milk in the first year of life. If you define fat as a general term for lipids and cholesterol there are many biological functions that require these to function. One of the most important of these is storing energy to be used later, as well as thermal insulation.
Calories are just a unit of measure an amount of energy.
In dietary circles calories are used as a measure of the amount of energy obtained from a certain type of food. Some foods are more energy rich than others. For example 8 oz of celery will contain less energy than 8 oz of chocolate. Therefore, we need energy for our bodies to function. Since the body tries to be efficient it will store the excess energy in the form of fat.