With increasing temperature a larger and larger fraction of the molecular bonds in a material will be excitated - thus energy is stored as potential and kinetic energy in their vibrations. This was first described by Einstein (http://en.wikipedia.org/wiki/Einstein_solid) and later further developed by Debye (http://en.wikipedia.org/wiki/Debye_model).

With increasing temperature a larger and larger fraction of the molecular bonds in a material will be excitated - thus energy is stored as potential and kinetic energy in their vibrations. This was first described by Einstein (http://en.wikipedia.org/wiki/Einstein_solid) and later further developed by Debye (http://en.wikipedia.org/wiki/Debye_model).

Mr. Thybrings explanation is fine. There are some books explaning the temperature dependence of the specific heat quite well. You can find a good explanation in the book of Charles Kittel, Introduction to Solid State Physics (chapters 4 and 5). This is online available in the net. In fact everything is related to the phonon spectrum inside a solid material. With increasing temperature higher vibration levels can be occupied, resulting in a increase in the specific heat. During structural changes (changes in the crystal structure) there might be a level change in the specific heat (up or down) as this has an impact on the phonon spectrum inside a material. Directly at a the first-order phase change the specific heat is not defined (I need to mention this to make sure that I will not be killed by my old thermodynamics professor). Around higher order phase transitions such as the ferromagentic-paramagnetic change in iron the specific heat shows anusual behavior as well.

As known that different amounts of energy are needed to raise the temperature of identical masses of different substance by one degree.  For example, 4.5 kJ of energy is needed to raise the temperature of iron form 20 to 30 deg. C while about 42 kJ (or 9 times of that energy) is needed to raise the temperature of 1 kg of liquid water from 20 to 30 deg. C.  It is therefore desirable to define a property to enable us to compare the energy storage capabilities to various substances, that is, the specific heat (referred to Thermodynamics-An Engineering Approach 6th. ed. by Cengel, Y.A. and Boles, M. A.)  

Quote from  the referred textbook Ch. 4 - the specific heat is defined as the energy required to raised the temperature of a unit mass of a substance by one degree.  In general, the energy depends on how the process is executed.  In thermodynamics, It is interested in two kinds of specific heats; specific heat at constant volume and specific heat at constant pressure. Unquote..