Latent Heat
Latent heat is the energy that is absorbed, then stored, and potentially released by a substance during a phase change. In meteorology, this has a fundamental application because water is able to absorb and release latent heat. The air in our atmosphere always holds water and in all three states - ice, liquid water and water vapour. This allows phase changes within the atmoshere and at the surface, as the temperature changes. Latent heat release in meteorology refers to the change of state of water vapour to water (condensation) and water to ice (freezing). During these phase changes, the molecules are moving into a more disordered state and inter-molecular bonds are broken. There is a net release of energy (heat) into the atmosphere as a result. 
     
The opposite of latent heat release can also occur, when ice changes to water (melting), or water changes to gas (evaporation). The change of water to gas has large implications in convective meteorology, and this process is known as evaporative cooling. In the atmosphere, most phase changes occur gradually over a range of different temperatures (and therefore heights): however, the phase change of ice to liquid occurs abruptly at the 0°C isotherm.   
This sounding from Norman, Oklahoma at 6pm on 20th July 2015 illustrates the dry adiabatic lapse rate and saturated adiabatic lapse rate nicely. Incidentally, this is a thunderstorm sounding, with almost 2000 J/kg of CAPE. The black line is the hypothetical path a parcel from the surface would take in the (likely) case that a thunderstorm would develop. 
The above example displays why latent heat release is so important for convective meteorology and thunderstorm development. In this case, a surface parcel is not quite convectively buoyant, so would need a push, such as a front or convergence of air, to enable the parcel to rise from the surface, whereby its temperature decreases at the dry adiabatic lapse rate (9.8°C/km). However, as the parcel rises, it cools towards its dewpoint, which in this case is around 22°C for a surface parcel. When the dewpoint is reached, the phase change from gas to liquid occurs and a cloud begins to form (at around 810 hPa) and latent heat release begins as the parcel continues to rise. Latent heat release warms up the air parcel, thereby essentially reducing the rate at which the parcel was cooling in the first place. So the saturated adiabatic lapse rate is reduced to around 6°C/km - in this case, the air parcel is finally made convectively buoyant, allowing around 2000J/kg of CAPE to be generated. If latent heat release did not occur, then the parcel would only ever cool at the dry adiabatic lapse rate (DALR) and thunderstorms would hardly ever occur. 
 
However, as can be seen by the curved saturated adiabatic lapse rate (SALR), this is not the whole story. The SALR is not constant - in fact it decreases with height. This is because the amount of water vapour that air can hold decreases with temperature and therefore with height. As the height increases through the atmosphere, the parcel holds less water vapour and so less latent heat release occurs, meaning that the lapse rate must increase as it is being less offset by this heat. Eventually, the parcel becomes so cold that latent heat release through condensation becomes negligable and the lapse rate becomes the DALR. There is also latent heat release of freezing, as the water droplets freeze above the 0°C isotherm, although this is also negligable and can be neglected.