"The Crown" means the 'government' of New Zealand.

Not in terms of Politicians/Parliament [the place where politicians hang out and carry on like they do], but in terms of things like law and order.

So, you might say the "public prosecutor (or defender?)" (or is it DA?), or is it "The people"? I can't be sure- too many US cop shows - I'm confused which is which.

New Zealand has a Westminster, Parliamentary style democracy, with the Queen of England [yep, good old Elizabeth the Second] as the nominal "head of state" [this arrangement dates back to the 1700 and 1800's when the country was colonised by England].

We have a (government of the day appointed (- for 5 years at a time) - with the Queens ok) Head of state called "The Governor General" who is the Head of State of the country (instead of the Queen), when the Queen is not actually in New Zealand [and The Queen is away a lot I've noticed too - I won't be voting for her anytime soon - too much absenteeism for my liking :-)) ]

Although the Queen of England/Governor General always defers any running of the country management stuff to the elected government of the day.
The Queen/Governor General can in theory "dissolve parliament" [If only getting rid of politicians was THAT easy] - which causes fresh elections to occur.

Thats never happened here, but did happen in Australia once in the early 1970s, and some say Australia has never been the same since because of it.

So, this connection with Royalty (i.e. a Crown) gives the "law and order" part of the government the name "The Crown".

So, when cases are tried in the courts for crimes like murder etc, its always "The Crown" who brings the case to the court, and who is responsible for administering justice to the offender once they are found guilty [like sticking them in jail] - we don't have a Death Sentence anymore since about 1957 or so.

We have had the occasional "private prosecution" for things like murder - but mostly its "The Crown" that brings the case to court.
[Bit like what OJ's ex-wife's familiy did to OJ - they bought a private prosecution for her murder].

I think the same applies in the UK - the law and order is represented in courts/in a legal sense by "The Crown".

Australia has a similar, but different enough legal system. I think they have a "Crown" in a legal sense and symbolic too, but Australia is a Federation of independent states (like the US of A is), so things get messy as you have "state" laws/and governments and "Federal" laws and governments.

Hope that makes things clear[er].