Relative humidity usually appears as a percentage, a nice method to difference it from other units of measure. For instance with a value of 80%, what's all about?
The most common effect might be in the form of heat perception due to the inherent system that the human organism has to dissipate temperature. Thanks to the sweat, by evaporation, the body releases enough temperature. Nonetheless any external condition that impedes the evaporation — like high humidity— causes a lower effectiveness.
Another possible effect has relationship with the chance to fog, rain or other atmospheric event where particles of water vapor come into action.
High humidity encourages the increase of volume in wood or other materials.
While moisture could affect the electric conduction of the air positively, air humidity seems to produce the inverse, so an object would tend to not conserve or produce static electricity.
In high humidity environments, at a certain temperature, the formation of microorganisms is done with more ease.
The respiration system and the organism in general prefers —perhaps— high humidity than a dry ambient since the natural barrier for infections may function in a different manner according to this factor. Coughs and other normalities are also not equal.
Without intention to cover every possibility as the list is enormous, simply to mention here another common possibility: how high humidity environments cause issues toward the bones and the necessity of precautions with special clothing and other measures. There is a discussion about the mechanism but not about the statistics.