
feucht
damp
Slightly wet or permeated by a small amount of liquid.
Wisch bitte den Tisch mit einem feuchten Tuch ab.
Please wipe the table with a damp cloth.
damp / humid
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damp
Slightly wet or permeated by a small amount of liquid.
Wisch bitte den Tisch mit einem feuchten Tuch ab.
Please wipe the table with a damp cloth.

humid / damp
Referring to weather or air: having a high concentration of water in the air; muggy or clammy.
Im Keller ist es immer etwas feucht und kühl.
It's always a bit damp and cool in the basement.

productive
Used in medicine to describe a cough in which mucus is produced and coughed up.
Bei einem feuchten Husten sollten Sie viel trinken, um den Schleim zu lösen.
With a productive cough, you should drink plenty of fluids to loosen the mucus.

to not care at all
(Colloquial, crude) To not be interested at all; to be completely indifferent.
Es interessiert mich einen feuchten Kehricht, was er über mich denkt.
I couldn't care less what he thinks about me.
The adjective 'feucht' (damp) belongs to the word family of 'die Feuchtigkeit' (moisture/humidity). When something is 'feucht' (damp), it contains a certain amount of moisture.
Die hohe Luftfeuchtigkeit im Bad führt zu feuchten Wänden.
The high humidity in the bathroom leads to damp walls.
In the basic meaning 'leicht nass' (slightly wet), 'feucht' (damp) is often combined with things that have absorbed water: damp hands, damp clothing, damp eyes, or a damp meadow.
Nach dem Spaziergang im Regen waren seine Socken ganz feucht.
After the walk in the rain, his socks were completely damp.
In connection with weather, climate, or air, 'feucht' (humid/damp) always means that the air has a high concentration of water. Typical combinations are 'feuchte Luft' (humid air) or 'feucht-warmes Wetter' (humid and warm weather).
An schwülen Sommertagen ist die Luft oft sehr feucht.
On muggy summer days, the air is often very humid.
In a medical context, almost exclusively in the combination 'feuchter Husten' (productive cough), the word describes a cough in which mucus is produced.
Der Arzt diagnostizierte einen feuchten Husten und verschrieb einen Hustenlöser.
The doctor diagnosed a productive cough and prescribed an expectorant.
The colloquial expression 'jemanden einen feuchten Kehricht/Dreck interessieren' means that someone is completely indifferent to something. It is very informal and expresses strong indifference.
Ob er mitkommt oder nicht, interessiert mich einen feuchten Kehricht.
Whether he comes along or not, I couldn't care less.
wet
Nass and feucht both describe something that contains liquid or has liquid on it. Feucht usually means that there is only a small amount; nass describes greater wetness, up to being completely soaked. The exact distinction depends on the context.

Nach einer Stunde auf der Heizung war das Handtuch nur noch feucht.
After an hour on the heater, the towel was only damp.
Direkt nach dem Schwimmen war das Handtuch nass und tropfte auf den Boden.
Right after swimming, the towel was wet and dripping onto the floor.