das Theaterstück
play
A literary work written to be performed on a stage by actors; a drama or a comedy.
Die Schulklasse probt gerade ein Theaterstück von Shakespeare.
The school class is currently rehearsing a play by Shakespeare.

play
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play
A literary work written to be performed on a stage by actors; a drama or a comedy.
Die Schulklasse probt gerade ein Theaterstück von Shakespeare.
The school class is currently rehearsing a play by Shakespeare.

'Theaterstück' (play) is a compound noun made up of 'das Theater' (the theater) and 'das Stück' (the piece/play). It literally means 'a piece for the theater'.
Dieses Stück ist perfekt für ein kleines Theater.
This play is perfect for a small theater.
In connection with creating and performing a play, certain verbs are very typical: to 'schreiben' (write), 'proben' (rehearse), 'inszenieren' (stage/direct), or 'aufführen' (perform) a play.
Die Theatergruppe wird das neue Theaterstück im Sommer aufführen.
The theater group will perform the new play in the summer.
From the audience's perspective, you use verbs like 'sich ein Theaterstück ansehen' (to watch a play) or 'in ein Theaterstück gehen' (to go to a play). Actors can also 'in einem Theaterstück mitspielen' (act in a play).
Wollen wir uns am Wochenende ein Theaterstück im Stadttheater ansehen?
Shall we go see a play at the city theater this weekend?
piece
'das Stück' and 'das Theaterstück' can mean the same thing in theater: a work that is performed on a stage. 'Theaterstück' is clearer because it is immediately obvious that it is about theater. 'Stück' is shorter and very common in conversation, but it has many other meanings. That is why it can easily be confused. If the theater context is clear, 'Stück' is often enough. If you want to avoid any possible confusion, 'Theaterstück' is more precise.

Im Deutschkurs lesen wir ein Theaterstück von Brecht.
In German class, we are reading a play by Brecht.
Nach der Probe spricht die Gruppe noch lange über das Stück.
After the rehearsal, the group talks about the play for a long time.