gemütlich
cozy
Describing a pleasant, relaxed, and comfortable atmosphere. Often refers to places or situations.
Wir saßen in einem gemütlichen Café und tranken heiße Schokolade.
We sat in a cozy café and drank hot chocolate.

cozy / relaxed
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cozy
Describing a pleasant, relaxed, and comfortable atmosphere. Often refers to places or situations.
Wir saßen in einem gemütlichen Café und tranken heiße Schokolade.
We sat in a cozy café and drank hot chocolate.

The word 'gemütlich' (cozy/comfortable) is derived from 'das Gemüt' (mind, disposition, or soul). Something 'Gemütliches' therefore appeals to inner well-being and creates a pleasant mood.
Die Atmosphäre in der Hütte war warm und gemütlich.
The atmosphere in the cabin was warm and cozy.
'Gemütlich' is very often used as an adverb to describe that an action is carried out in a slow, relaxed, and pleasant way.
Am Sonntag frühstücken wir immer ganz gemütlich auf dem Balkon.
On Sundays, we always have breakfast very leisurely on the balcony.
Common expressions are 'ein gemütlicher Abend' (a cozy evening), 'eine gemütliche Runde' (a cozy get-together), or the reflexive construction 'es sich gemütlich machen', which means to make yourself comfortable and relax.
Mach es dir doch auf dem Sofa gemütlich!
Make yourself comfortable on the sofa!
When 'gemütlich' describes an activity, it is often about a slow pace. Typical examples are 'ein gemütlicher Spaziergang' (a relaxed walk), 'eine gemütliche Radtour' (a relaxed bike ride), or 'etwas gemütlich angehen' (to take something at an easy pace).
Lass uns die Arbeit morgen gemütlich angehen.
Let's take the work at an easy pace tomorrow.
slow
'langsam' and 'gemütlich' can both describe a calm pace. 'langsam' mainly says that something is not fast. This can be neutral, or it can sound negative if something takes too long. 'gemütlich', on the other hand, means a slow pace that is pleasant, relaxed, and stress-free. That is why 'gemütlich' is more suitable for walks, meals, or free-time activities. Learners may confuse the words because both have to do with low speed. The difference is: 'langsam' describes the speed, while 'gemütlich' describes the relaxed manner.

Nach dem Mittagessen gehen wir gemütlich am Fluss entlang.
After lunch, we take a relaxed walk along the river.
Der Bus fährt heute sehr langsam durch die Innenstadt.
The bus is driving very slowly through the city center today.