die Hälfte
half
One of two equal parts of a whole.
Ich hätte gern die Hälfte des Kuchens, bitte.
I’d like half of the cake, please.

half
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half
One of two equal parts of a whole.
Ich hätte gern die Hälfte des Kuchens, bitte.
I’d like half of the cake, please.

The noun 'die Hälfte' (half) is derived directly from the adjective 'halb' (half). It literally describes the state of something having been divided in half.
Ein halber Apfel ist die Hälfte eines ganzen Apfels.
Half an apple is half of a whole apple.
'Hälfte' (half) is very often used with the genitive to say what it is half of. In colloquial speech, 'von' + dative is often used instead.
Die Hälfte der Bevölkerung lebt in Städten.
Half of the population lives in cities.
The phrase 'um die Hälfte' (by half) is used to describe a reduction or increase of 50%.
Die Firma hat ihre Kosten um die Hälfte reduziert.
The company reduced its costs by half.
The phrase 'zur Hälfte' (half / halfway) describes that something is only partially or 50% finished or filled.
Das Stadion war zur Hälfte leer.
The stadium was half empty.
The expression 'bessere Hälfte' (better half) for one’s partner is informal and is often used in an affectionate, ironic way.
Ich muss erst meine bessere Hälfte fragen, ob wir am Samstag Zeit haben.
I first have to ask my better half whether we have time on Saturday.
part
'der Teil' and 'die Hälfte' both have to do with a whole. 'der Teil' is general: a part can be large or small, and there can be many parts. 'die Hälfte' is more specific: the whole is divided into two equal parts. So every half is a part, but not every part is a half. Learners may confuse the words when they only want to say that something is not whole. If the size is exactly 50 percent, you say 'Hälfte'; otherwise, 'Teil' is safer.

Die Hälfte der Wohnung ist schon gestrichen.
Half of the apartment has already been painted.
Sie hat nur einen kleinen Teil der Rechnung bezahlt.
She paid only a small part of the bill.