Calorimetry is simply the measurement of heat. This is typically done under two distinct conditions.
Either at constant pressure in which case
\[q=q_p=\Delta H\]
Or at constant volume in which case
\[q=q_v=\Delta U\]
The two instruments we typically discuss for these situations are: a coffee cup calorimeter (constant pressure calorimeter), and a bomb calorimeter (constant volume calorimeter).
In a calorimeter we take advantage of the fact that we can use the temperature change of water to quantify an amount of heat. A typical calorimeter works by simply capturing all the energy released (or absorbed) by a reaction in a water bath. The energy that enters (leaves) the water must be exactly equal to the energy released (absorbed) by the chemical reaction. Thus by measuring the change in the temperature of the water we can quantify the heat (enthalpy) of the chemical reaction.
Attached below is a helpful sheet on calorimetry from Dr. Paul McCord. When working calorimetry problems it is important to read the problem to see how the calorimeter is calibrated. Sometimes there is just the heat capacity of the calorimeter, other times you have to take into account the specific heat and mass of the water or solution in the calorimeter and the heat capacity of the hardware of the calorimeter (together they make the heat capacity of the calorimeter) and yet other times you can assume the heat capacity of the hardware to be zero (as is often the case with coffee-cup calorimetry BUT NOT ALWAYS!) and you just have to consider the specific heat and mass of the water in the calorimeter.
Here is Dr. McCord's Calorimetry Help Sheet