das Gefühl
feeling
An inner, emotional feeling such as joy, sadness, anger, or fear.
Beim Abschied überkam sie ein Gefühl der Traurigkeit.
At the farewell, she was overcome by a feeling of sadness.

feeling / sense
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feeling
An inner, emotional feeling such as joy, sadness, anger, or fear.
Beim Abschied überkam sie ein Gefühl der Traurigkeit.
At the farewell, she was overcome by a feeling of sadness.

The noun 'das Gefühl' (the feeling/sense) is derived directly from the verb 'fühlen' (to feel). This connection helps you understand the basic meaning of inner or outer perception.
Ich fühle eine große Freude, und dieses Gefühl ist wunderbar.
I feel great joy, and this feeling is wonderful.
When 'Gefühl' (feeling) describes a hunch or intuition, it is often followed by a subordinate clause with 'dass' (that) or an infinitive construction with 'zu' (to). The fixed phrase is 'das Gefühl haben, dass/zu ...' (to have the feeling that/to ...).
Ich habe das Gefühl, dass ich das schon einmal erlebt habe.
I have the feeling that I have experienced this before.
To express a talent or sense for something, you use the construction 'ein Gefühl für etwas haben' (to have a sense for something). The thing someone has a sense for is always in the accusative.
Ein guter Musiker hat ein feines Gefühl für den Rhythmus.
A good musician has a fine sense of rhythm.
In an emotional context, 'Gefühl' (feeling) is often combined with verbs such as 'zeigen' (to show), 'verbergen' (to hide), or 'verletzen' (to hurt). The combinations 'ein gutes/schlechtes Gefühl' (a good/bad feeling) are also very common.
Er versucht, seine wahren Gefühle nicht zu zeigen.
He tries not to show his true feelings.
You can recognize the meaning 'Tastsinn' (sense of touch) from the context. It is almost always used in connection with body parts and often in negative sentences to describe a loss of sensation.
Wegen der Kälte hatte sie kein Gefühl mehr in den Fingern.
Because of the cold, she no longer had any feeling in her fingers.
meaning / sense
'das Gefühl' and 'der Sinn' can both refer to perception or a special feel for something. 'Gefühl' often means an inner feeling, but it can also mean physical sensation or a talent, for example 'ein Gefühl für Sprache' (a feel for language). In the context of perception, 'Sinn' more often means a fixed ability such as seeing, hearing, or feeling; in the expression 'Sinn für etwas haben' (to have a sense for something), it also means a good understanding or feel for something. Learners may confuse the words because you can say both 'Gefühl für Musik' and 'Sinn für Musik'. 'Gefühl' often sounds more personal and intuitive, while 'Sinn' sounds more like an ability or understanding.

Nach dem langen Spaziergang hatte Jan kaum noch Gefühl in den Fingern.
After the long walk, Jan could hardly feel his fingers anymore.
Für Farben hat Sofia einen guten Sinn und richtet ihre Wohnung sehr schön ein.
Sofia has a good sense for colors and decorates her apartment very beautifully.