Wet is the highest degree of wetness. Moist, damp and humid are not as wet. When something is soaked in and is dripping of liquid, we call it wet. A swab well soaked with water is wet.
Moist and damp have more or less the same degree of wetness, but moist is used mainly in places where the wetness is desirable, i.e. it is a positive word. Damp on the other hand is used when wetness is undesired, i.e it is a negative word.
For example, if you play football (English - football/American - soccer) you will understand that a moist field is good, but a damp field is not. Cake and bread must be moist, but not damp. Clothes that aren't dry enough to be worn are damp, not moist. The mud by the river is damp.
Humidity is connected with atmosphere/air. Air has moisture in it and it causes humidity, too much of which makes one feel damp.