Having a degree with a major in Government (BA Syd), also having lived in NSW for 65 years and worked as a polling officer, then lived in NZ for 9 years and being able to vote for 8 (5 before I gained citizenship) of those years, I am well aware of the differences.
NZ has a unicameral system and, of course, no state governments. There are 120 members. There are 64 general electorates and 7 Maori electorates. Each person is allowed to choose (I think every 5 years) as to in which electorate they will vote.
When you vote, you merely put 2 ticks. I tick is for the member in your electorate (First Past the Post) and the other tick is for a party.
The 120 members are decided by the party vote. After electorate members are decided (obviously much quicker than Australia due to being first past the post), the remaining 49 members are allocated according to the party vote from lists provided by the parties before the election so that the total number of party members are in accordance with its % of the party vote.
A party is only represented in parliament if it either wins an electorate or gains 5% or more of the party vote. This is called MMP (Mixed Member Proportional) and a similar system is used in Germany.
I think this system is why we do not tend to get extremists on either end of the spectrum in the NZ parliament.
Personally I vote Labour in my electorate as there is little point in voting for the Green candidate with First Past the Post, it would only help the conservative candidate. I party vote Green.
Voting is not compulsory in NZ but enrolling is and you must enrol once you have been a resident for 1 year.