Calories are therefore units of heat energy and in the context of nutrition they represent a food's ability to provide useful energy for human metabolism, muscular contraction (movement) and, when more calories are consumed than needed for these activities, for the production of adipose tissue (body fat).
It is important to understand that calories are not fat, carbohydrate or any other kind of nutrient, rather they are the latent energy contained within those nutrients, specifically the energy that is released when the chemical bonds of those nutrients are broken.
To better grasp this, consider a pile of wood. That wood is largely composed of a carbohydrate called cellulose and the chemical bonds in the wood's cellulose molecules contain energy that can be liberated when those bonds are broken. In order for us to percieve and feel this energy all we have to do is light the wood pile on fire. All the heat that is generated by that fire is coming from the breakdown of the chemical bonds in cellulose.
Sometimes the body tends to just "burn" food calories into heat in much the same way as a fire, but more often, it uses the calories to perform work in much the same way that a steam engine can capture some heat from a fire and move a train. When all the essential work of the body is tended to, any left over calories are used to power fat storage in adipose tissue.