Heat Capacity

Heat Capacity
When a substance is heated, its change in temperature(∆T) is proportional to the quantity of heat energy it absorbs(q), thus:

q/∆T = constant

The value of this constant is different for each substance and is called that substance's 'heat capacity'.

Specific Heat Capacity
The specific heat capacity(c) of a substance is the amount of heat energy required to raise 1 gram of that substance by 1 degree kelvin.

c = q/(mass * ∆T)

Molar Heat Capacity
The molar heat capacity(C) of a substance is the amount of heat energy required to raise the temperature of 1 mole of that substance by 1 degree kelvin.

C = q/(moles * ∆T)

Units
Heat energy is in joules, temperature is in degrees kelvin, and mass is in grams, thus c has units of J/(K*g) and C has units of J/(K*mol).

Enthalpy

(See Energy first.)

What is enthalpy?
Enthalpy(H), or 'heat content,' is a variable that describes the thermodynamic potential(hence 'heat content') of a system.
The enthalpy of a system is defined as its internal energy plus the product of the pressure on it(P) and its volume(V).

H = E + PV

How is it used?
In many reactions the only type of work done is that of a newly formed gas expanding and pushing back the atmosphere. This is called PV work, and is equal to the product of the pressure of the atmosphere and the change in volume of the gas.

w = -P∆V

The value is negative because the gas(system) is doing work on, and thus losing energy to, the atmosphere(surroundings).

In reactions such as these, if the pressure is constant (as is most common) it is often more meaningful and convenient to examine the change in enthalpy of a system rather than the change in its internal energy. This is because by using enthalpy we can avoid having to consider PV (and thus all) work done by the system.
This is how:

H = E + PV

so at constant pressure

∆H = ∆E + P∆V

and since

∆E = q + w
&
w = -P∆V

we can see that

∆E = q - P∆V
& rearranging
q = ∆E + P∆V

so

∆H = q

As this is only true at constant pressure we write the above as

∆H = q_p

with _p effectively meaning 'at constant pressure'.

Thus to find the change in enthalpy of a system under these conditions, we only have to measure the heat transferred between the system and its surroundings.

Further
For reactions that involve little or no work, and thus where ∆E = q, knowing ∆H means knowing ∆E.