The heat capacity specifies the heat needed to raise a certain amount of a substance by 1 K. For a gas, the molar heat capacity C is the heat required to increase the temperature of 1 mole of gas by 1 K.
Defining statement: dQ = nC dT
Important: The heat capacity depends on whether the heat is added at constant volume or constant pressure.
Why care about the difference?
Defining statement: dQ = nC_{V} dT
1st Law of Thermodynamics gives: dQ = dE  dW = dE at constant volume.
Thus dQ = dE = nC_{V} dT = 

nR dT , 
Comparing, we get, for a monatomic ideal gas: C_{V}  = 

R  (diatomic:  C_{V}  = 

R) 
Since dQ = dE when the volume is
constant,
dE = n C_{V} dT.