The glass is in a non-equilibrium state in nature, and relaxation might occur towards the equilibrium state at a certain temperature. When heating a quenched glass, relaxation can be resolved as temperature approaches to the glass transition, and further heating leads to enthalpy recovery as the system turns into an equilibrium supercooled liquid. The released energy involving the relaxation relative to the original quenched state is, in magnitude, identical to the gained energy in enthalpy recovery, showing a memory effect. In this paper, we discuss the enthalpy behaviors involved in a cooling and reheating cycle around the glass transition in various glass forming systems such as oxides, metal alloys, and small molecular systems. The cooling and heating rates are fixed to be -/+ 20 K/min with the related cooling and heating heat capacity curves being determined. It is found that the relaxation enthalpy involved in the cooling/heating cycles is closely related to the enthalpy of fusion for the glass forming materials, and the basically linear correlation implies the similarity between the glass transition and melting behaviors with regard to the atomic rearrangements involved in the relaxation and solidification processes. The determining of the cooling and heating heat capacity curves also helps establish the enthalpy relaxation/recovery spectra of various glasses, and the symmetry of the spectrum is associated with the fragility of glass-forming material. For the material of low or medium fragilities, the symmetry of the enthalpy relaxation spectrum is observed to be somehow dependent on the fragility, while for the high fragility glass, the symmetry keeps almost constant. The dependence of fragility on the glass transition thermodynamics is also discussed, and low melting entropy and high fragility are shown to reduce effectively the liquid-crystal Gibbs free energy difference. Using the correlation between the relaxation enthalpy and kinetic fragility reported in our previous studies, the glass transition thermodynamics for the case of the most fragile glass with m= 175 is evaluated, especially compared with the second phase transition of thermodynamics. The results provide a new understanding of the thermodynamics of the relaxation in glassy material and the glass transition.