Do I understand why heat capacity is called 'heat *capacity*'?
Expanding on Nicolau Saker Neto's comments above, the heat capacity dictates the (reversible) heat flux required to increase the system's temperature by one unit. 

Definitions of the heat capacity ($C_V$ and $C_p$)

Another way to look at it is that the heat capacity dictates the response in internal energy ($U$)/enthalpy ($H$) of the system to temperature, as described below. Note that the heat capacity at constant volume ($C_V$) is not the same as the heat capacity at constant pressure ($C_p$) - and that both may be defined in several ways, the most common being: 

$$ C_V = \left(\frac{\partial U}{\partial T}\right)_{V,\mathbf{n}} = \left(\frac{\partial Q}{\partial T}\right)_{V,\mathbf{n}} $$

$$ C_p = \left(\frac{\partial H}{\partial T}\right)_{p,\mathbf{n}} = \left(\frac{\partial Q}{\partial T}\right)_{p,\mathbf{n}} $$

Contributions to the heat capacity

In should be added that while translation - the kinetic energy of molecules in motion - are an important contribution to the heat capacity of a gas, both vibrations and rotations also contribute for all but the simplest molecules. For high temperatures (approx. $T \geq 3000\ \mathrm{K}) $, the excitation of outer electrons will also contribute to the heat capacity. For even higher temperatures, the substance in question will behave as plasma. 

In short

There are many complicating factors to the precise understanding of the origin of contributions to the heat capacity - both at the high end and low end of the temperature range. In all cases, however, the heat capacity will give you the answer to the question: 


  "How much energy do I have to add to this system at the given
  conditions in order to increase the temperature with one unit?"


Further reading on Wikipedia is recommended for anyone interested.