der Daumen
thumb
The first, shortest, and strongest finger of the human hand.
Er hat sich beim Hämmern auf den Daumen geschlagen.
He hit his thumb while hammering.

thumb
Loading your saved state.
thumb
The first, shortest, and strongest finger of the human hand.
Er hat sich beim Hämmern auf den Daumen geschlagen.
He hit his thumb while hammering.

The idiom 'jemandem die Daumen drücken' (to keep one's fingers crossed for someone) means to wish someone luck. The person you wish luck to is always in the dative.
Ich drücke dir für das Vorstellungsgespräch die Daumen.
I'll keep my fingers crossed for your job interview.
The expression 'über den Daumen peilen' (to estimate roughly) means to estimate something only roughly. The participle form 'über den Daumen gepeilt' (roughly estimated) is also often used at the beginning of a sentence.
Über den Daumen gepeilt, brauchen wir für die Fahrt zwei Stunden.
Roughly estimated, we'll need two hours for the drive.
The gesture 'Daumen hoch' (thumbs up) is a very common positive signal in Germany. It means 'great', 'all right', or 'well done'.
Als ich fragte, ob alles okay sei, zeigte er mir nur den Daumen.
When I asked whether everything was okay, he just gave me a thumbs-up.
finger
'der Finger' and 'der Daumen' both belong to the hand. The difference is: 'Finger' usually means one of the movable parts of the hand, not including the thumb, for example the index finger or middle finger. 'Der Daumen' is the short, strong finger on the side of the hand. In a general sense, you can also see the thumb as a finger, but in everyday language it is almost always called 'Daumen' separately. Learners may confuse the words because the thumb anatomically belongs to the hand like the fingers. If you want to be precise, use 'Daumen' for this part of the hand.

Max hebt den Daumen, weil ihm die Idee gefällt.
Max gives a thumbs-up because he likes the idea.
Beim Schneiden habe ich mir in den Finger geschnitten.
I cut my finger while chopping.