die Puppe
doll
A toy figure that represents a human being, often a baby or child, or a human-like figure.
Zum Geburtstag wünscht sich das Kind eine sprechende Puppe.
For their birthday, the child wants a talking doll.

doll / mannequin / pupa
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doll
A toy figure that represents a human being, often a baby or child, or a human-like figure.
Zum Geburtstag wünscht sich das Kind eine sprechende Puppe.
For their birthday, the child wants a talking doll.

The idiom 'die Puppen tanzen lassen' literally means 'to let the dolls dance' and means to celebrate exuberantly, throw a big party, or cause a lot of excitement.
Auf seiner Geburtstagsfeier haben wir so richtig die Puppen tanzen lassen.
At his birthday party, we really let loose.
When referring to the toy, typical verbs and nouns are 'mit Puppen spielen' (to play with dolls), 'eine Puppe anziehen' (to dress a doll), 'Puppenhaus' (dollhouse), or 'Puppenwagen' (doll stroller/pram).
Als Kind habe ich stundenlang mit meinen Puppen gespielt.
As a child, I played with my dolls for hours.
When 'Puppe' (doll/mannequin) is used in the context of clothing, fashion, a store, or a 'Schaufenster' (shop window), it means a mannequin.
Der Mantel sieht an der Puppe im Schaufenster toll aus.
The coat looks great on the mannequin in the shop window.
The biological meaning can be recognized from words like 'Insekt' (insect), 'Schmetterling' (butterfly), 'Larve' (larva), or the verb 'sich verpuppen' (to pupate). In this context, it is a technical term.
Die Raupe hat sich in eine Puppe verwandelt.
The caterpillar has transformed into a pupa.
As a form of address or term for a girl or woman, 'Puppe' (doll) is colloquial and usually meant affectionately or familiarly. It is often used with a possessive pronoun like 'meine' (my).
Hallo meine Puppe, wie war dein Tag in der Schule?
Hello, my doll, how was your day at school?
girl
'das Mädchen' and 'die Puppe' can both refer to a little girl, but only in a special usage. 'Mädchen' is neutral and fits in almost any normal situation. 'Puppe' is not actually a neutral word for a girl; rather, it can be colloquial and meant affectionately, especially for small children. However, it can also easily sound too personal or old-fashioned. In everyday language, 'Mädchen' is usually safer.

Die Oma nennt ihre Enkelin liebevoll ihre kleine Puppe.
The grandma affectionately calls her granddaughter her little doll.
Das Mädchen hat heute im Park mit anderen Kindern gespielt.
The girl played with other children in the park today.