Specific heat capacity relates to pure compounds. I.e. if I have 1g of pure water, it takes 4.18J of energy to increase the temperature by 1 degree. If I have 2g, it will then take 8.36J to heat it by 1 degree. If I have 2g and I want to increase the temperature by 10 degrees, I will need 83.6J of energy. And so forth. The unit is thus J g-1 C-1.
Calibration factor is the 'heat capacity' of a calorimeter, i.e. how much it takes to heat up the entire calorimeter by 1 degree. So in here, you will have some water, a thermometer, a metallic cup, some polystyrene, etc. Each object has its own specific heat capacity, and each object has a certain mass. So we can calculate how much energy it takes to heat the water, and the glass in the thermometer, and so forth, and add up all the heat capacities to give us an overall value that tell us how much energy is required to heat everything up by 1 degree. This is the calibration factor. However, since measuring the mass of each component can be tedious and may not always be possible, we therefore do a simple calibration experiment to determine this calibration factor.