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.

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.
hand
'die Hand' and 'die Klaue' can both refer to a human hand, but they sound very different. 'Hand' is the normal, neutral word for the body part. 'Klaue' actually means the sharp nail or foot of an animal. Colloquially, you can also use 'Klaue' for a human hand, but it often sounds rough or derogatory. In addition, 'Klaue' can mean very illegible handwriting. The words can be confused because both sometimes have to do with a hand or handwriting. In normal situations, you almost always say 'Hand'.

Deine Klaue kann ich kaum lesen, schreib bitte etwas ordentlicher.
I can barely read your awful handwriting; please write a bit more neatly.
Nach dem langen Schreiben tut mir die Hand weh.
After writing for so long, my hand hurts.