Let's go back to the first law of thermodynamics

ΔU = q + w

and ask ourselves the question "If energy is so important a concept, then how are we going to measure it?" There are no energy meters per se and I'll give you a hint. You will never see an experimental measurement of U and so trying to find ΔU by measuring actual values of U directly at the beginning and end of a reaction or phase transition (or whatever chemical process you are trying to follow) and subtracting the two

ΔU = U_end - U_start

is unfortunately not an option. So we will measure the heat and work separately and add them together to get the change in energy. If we carry out our change isochorically (V = constant), then no "PV" work is done and the first law yields a particularly simple equation

ΔU = q_v + C_vΔT

In other words, at constant volume the energy change is equal to the heat transferred, and we can determine the energy by measuring the temperature. The subscript "V" indicates that the quantities were determined at constant volume.

C_v is the heat capacity at constant volume. The heat capacity is defined as the rate at which the energy of a given chemical increases with temperature and in differential form it looks like this

While working at constant volume is convenient for some applications, many changes that we want to study occur at constant pressure - that is at ambient laboratory conditions in open containers. This is exactly why enthalpy has been defined as

H = U + PV.

At constant pressure

the change in enthalpy is equal to the heat. We can again define a heat capacity at constant pressure to allow us to measure energy changes (as enthalpy)

Again, temperature is all we need to measure this energy change. In differential form, the heat capacity at constant pressure looks like this

Because chemists often work on laboratory bench tops in open beakers and test tubes, constant pressure (isobaric) measurements are very common.

Heat capacities are extrinsic properties of chemical systems; they depend upon the amount of substance you have. If you wanted to heat some water for tea, you would have to put twice as much energy per degree K into heating two cups of water as you would one cup. So we will have to match the units of heat capacity to reflect the amount of material we want to heat. We can make the heat capacity into an intrinsic property (independent of the amount present) by expressing it per unit amount. If we do this per mole, the molar heat capacity C_p will have units of Joules/K-mole; if we do this per gram, the specific heat C_swill have units of Joules/K-gram. Be sure to match your units and amounts of substance to that of the heat capacity. Here is a units converter to help you do that.

For a more descriptive treatment of heat capacities try this web page. Be sure to check out the link A Home-Cooked Experiment in Heat Capacity