The molar heat capacity tells you how many joules (or calories) of heat flow into a substance are necessary to raise a mole of the substance one degree Kelvin (or Celsius). That's just dimensional analysis. At a deeper level, the molar heat capacity of a substance tells you about how many ways that the particles of a substance can move to hold kinetic energy. In general, the more ways the molecules of a substance can move, whether in a line, rotating, or vibrating, the greater the molar heat capacity is going to be. Each translational, vibrational and rotational mode is a partition into which thermal energy can be distributed. The more places a substance has to put the energy, the greater the molar heat capacity.
Picture an imaginary merry-go-round on wheels, a vehicle which can hold kinetic energy not only in translational motion but also in rotation and the oscillations of the horses up and down. If a certain amount of kinetic energy were imparted on this vehicle, the energy could be distributed throughout all these 'partitions', some would increase its translational motion, but there is also room for energy in the rotations and vibrations. Because there are so many places to put the energy, it would take a lot of energy before any one partition would become vigorous.
Now picture a diatomic molecule, such as O2. In the terminology of kinetic theory, a diatomic molecule possesses more degrees of freedom than a single atom. O2 can have vibrational and rotational motions as well as translational motion. Heat flow into the sample causes the average translational energy to increase, but it also causes the energy associated with vibrational and rotational motion to increase. The oxygen gas will require a higher energy input to change the temperature by a certain amount. It would take a lot of energy before any one partition would become vigorous. In other words, O2 has a higher molar heat capacity than a monatomic gas such as helium.