fettig
greasy
Covered, soaked, or smeared with fat or oil.
Nach dem Essen der Pommes hatte ich ganz fettige Finger.
After eating the fries, my fingers were completely greasy.

greasy / oily
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greasy
Covered, soaked, or smeared with fat or oil.
Nach dem Essen der Pommes hatte ich ganz fettige Finger.
After eating the fries, my fingers were completely greasy.

The adjective 'fettig' (greasy/oily) is derived directly from the noun 'das Fett' (fat). The ending '-ig' turns the noun into an adjective and means 'having the quality of something.'
Die Soße war mir etwas zu fettig.
The sauce was a bit too greasy for me.
Do not confuse 'fettig' (greasy/oily) with 'fett' (fat). A person or animal can be 'fett' (overweight), but their skin or hair is 'fettig' (oily). 'Fettig' describes an oily surface.
Meine Haut im Gesicht ist leider sehr fettig.
Unfortunately, the skin on my face is very oily.
In connection with personal care, 'fettig' (oily) is almost always used for 'fettige Haut' (oily skin) and 'fettige Haare' (oily hair). It describes a condition that many people want to avoid.
Es gibt spezielle Shampoos für fettiges Haar.
There are special shampoos for oily hair.
When food is described as 'fettig' (greasy), it is often informal and has a slightly negative meaning. It implies that the food is very rich or perhaps even unhealthy.
Das Essen in der Kantine ist oft ziemlich fettig.
The food in the cafeteria is often pretty greasy.
fat / bold
'fett' and 'fettig' can both be used for food, but they emphasize different things. 'fett' means that something contains a lot of fat, for example cheese, meat, or cream. 'fettig' means that something seems oily or covered with fat. It often sounds more negative, especially with food, hands, skin, or hair. Learners easily confuse the words because both have to do with fat. As a guide: 'fett' describes the fat content, while 'fettig' is more about the oily surface or the unpleasant impression.

Nach dem Reparieren waren Toms Hände ganz fettig.
After doing the repairs, Tom's hands were completely greasy.
Diese Milch ist sehr fett und macht den Kaffee besonders cremig.
This milk is very fatty and makes the coffee especially creamy.