What is enthalpy and how can it be calculated?

Enthalpy is a thermodynamic property that chemists can use to describe the heat energy transferred throughout a chemical reaction. It is especially useful given that it is defined as "a measure of heat energy present in a chemical system at constant pressure". Interestingly, it is the constant pressure aspect of the definition that makes it so useful as most reactions - at least certainly the more straightforward reactions - can be conducted under such constant pressure conditions.
As an example, we can consider the bond strengths of individual molecules. Comparing the reactants to the products bond strengths give a numerical value of enthalpy that allows us to compare between different reactions.
CH₄ + 2O₂ --> CO₂ + 2H₂O [complete combustion of methane]
C-H bond enthalpy (here as a measure of bond strength) = 411 kJ mol-1O=O bond enthalpy = 494 kJ mol-1C=O bond enthalpy = 799 kJ mol-1O-H bond enthalpy = 459 kJ mol-1
Therefore, calculation to conduct (Reactants - Products) for enthalpy change of combustion = [4(411) + 2(494)] - [2(799) + 4(459)] = - 802 kJ mol-1 [exothermic]