die Handschrift
handwriting
The personal and characteristic way a person writes by hand.
Ich kann seine Handschrift kaum entziffern, so unleserlich ist sie.
I can hardly decipher his handwriting; it's so illegible.

handwriting / manuscript
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handwriting
The personal and characteristic way a person writes by hand.
Ich kann seine Handschrift kaum entziffern, so unleserlich ist sie.
I can hardly decipher his handwriting; it's so illegible.

The word is made up of 'Hand' (hand) and 'Schrift' (writing/script). So it literally means 'writing by hand.' This origin explains both the personal way of writing and old handwritten documents.
Seine Handschrift ist sehr elegant und gut zu lesen.
His handwriting is very elegant and easy to read.
Adjectives help distinguish the meanings: 'leserlich' (legible) or 'unordentlich' (messy) describes the way of writing. 'Persönlich' (personal) or 'unverkennbar' (unmistakable) points to an artist's style. 'Mittelalterlich' (medieval) or 'alt' (old) indicates a historical document.
Trotz seines Alters hat der Professor noch eine sehr saubere Handschrift.
Despite his age, the professor still has very neat handwriting.
When referring to the personal way of writing, 'Handschrift' (handwriting) is often combined with verbs such as 'haben' (to have), 'erkennen' (to recognize), or 'entziffern' (to decipher). Typical adjectives are 'schön' (beautiful), 'schlecht' (bad), 'leserlich' (legible), or 'unleserlich' (illegible).
Ich konnte die Notizen nicht lesen, weil ich seine Handschrift nicht entziffern kann.
I couldn't read the notes because I can't decipher his handwriting.
To describe the style of an artist or author, the fixed expression 'die Handschrift von jemandem tragen' (to bear someone's signature style) is often used. This emphasizes the unique and recognizable character of a work.
Dieses Gebäude trägt klar die Handschrift des berühmten Architekten.
This building clearly bears the signature style of the famous architect.
hand
'die Hand' and 'die Handschrift' can both mean the personal way someone writes. In this sense, they are similar. However, 'die Handschrift' is the normal and clear word for this. 'die Hand' in this meaning sounds more specialized and tends to occur in fixed or older expressions, for example with old letters or documents. They can be confused because 'Hand' usually means the body part. If you are simply talking about legible or illegible writing, 'Handschrift' is usually more natural.

Deine Handschrift ist heute viel leichter zu lesen als gestern.
Your handwriting is much easier to read today than it was yesterday.
Der alte Brief ist in einer sehr feinen Hand geschrieben.
The old letter is written in a very fine hand.