Direct calorimetry is a direct measure of heat production, whereas indirect calorimetry is a measure of O2 consumption and CO2 production.
Direct calorimetry is usually performed using a bomb calorimeter, where food is burned in a sealed container and the amount of heat produced is then converted into the number of calories that the food contains. The same process can be used to determine someone metabolic activities. By having someone sit in a sealed chamber and determine the amount of heat produced by their body. The heat produced can be converted to the number of calories burned.
Indirect calorimetry can be applied to individuals as well. For example, by having someone exercise while attached to a metabolic cart, you can measure their exact oxygen consumption and carbon dioxide production. You can then measure how much oxygen the individual used as fuel. This calculation should give an accurate representation of their energy expenditure.
Indirect calorimetry is used much more often than direct calorimetry in terms of determining energy expenditure for individuals because it is much cheeper and easier to administer.