
die Klaue
claw
The sharp, curved horn nail on the foot of predatory animals or birds of prey.
Der Falke packte die Maus mit seinen spitzen Klauen.
The falcon grabbed the mouse with its sharp claws.
claw / hoof
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claw
The sharp, curved horn nail on the foot of predatory animals or birds of prey.
Der Falke packte die Maus mit seinen spitzen Klauen.
The falcon grabbed the mouse with its sharp claws.

paw / mitt
Colloquial and often derogatory for a person's hand.
Nimm deine dreckigen Klauen von meinem Handy!
Get your dirty paws off my phone!

awful handwriting
Colloquial for illegible, scrawly handwriting.
Ich kann deine Notizen nicht lesen, du hast eine furchtbare Klaue.
I can't read your notes; you have awful handwriting.

hoof
In zoology, the horn capsule that encloses the tips of the toes in even-toed ungulates, such as cattle, sheep, and pigs.
Der Landwirt muss regelmäßig die Klauen seiner Kühe pflegen lassen.
The farmer has to have his cows' hooves cared for regularly.
The type of animal reveals the exact meaning. With predators and birds, such as eagles or cats, it means the sharp claws used for gripping. With farm animals, such as cows or sheep, it means hooves.
Die Katze fuhr ihre Klauen aus, als der Hund zu nahe kam.
The cat extended its claws when the dog came too close.
When 'Klaue' is used in connection with people, it is always colloquial. 'Eine Klaue haben' refers to illegible handwriting, while the plural 'Klauen' is a derogatory word for hands.
Entschuldigung für meine Klaue, ich habe schnell geschrieben.
Sorry for my awful handwriting; I was writing quickly.
Using 'Klaue' for a human hand is very colloquial and has a negative, often aggressive or derogatory connotation.
Hände weg von meinem Essen, du mit deinen Klauen!
Hands off my food, you and your paws!
The meaning 'bad handwriting' is almost always used in the fixed expression 'eine Klaue haben' (to have awful handwriting). The adjective before it can describe the degree.
Mein Arzt hat so eine schreckliche Klaue, ich kann das Rezept nicht lesen.
My doctor has such terrible handwriting that I can't read the prescription.
claw
Both words can refer to the sharp, curved claw of an animal. Kralle is the more general term for predators, birds, and reptiles, while Klaue particularly suggests a powerful, curved claw of a predator or bird of prey. With such animals, both words are sometimes possible, but the collocation determines the choice.

Der Adler schlug seine kräftige Klaue in die Beute und hielt sie fest.
The eagle drove its powerful claw into its prey and held it tightly.
Der Adler schloss seine Kralle um den Ast und blieb sicher sitzen.
The eagle closed its claw around the branch and remained securely perched.