In the specific context you're asking about these refer to qualifications at different levels of the New Zealand Qualifications Framework, which are documented in detail (PDF). The short answer, though, is to pick the name of the qualification that matches what you're planning to enrol in. The institution will list it with one of those terms specifically; if you don't know what you're going to enrol in yet, the first thing to do is figure that out.
The two "Postgraduate X" qualifications are the least widespread terms and do have some New Zealand-specific subtleties in terms of how they're offered. A Postgraduate Certificate requires 60 credits at level 8, and represents half a year of full-time tertiary study at an advanced level. A Postgraduate Diploma is 120 credits of same, representing a full year. These are defined on pages 17 & 18. Note that these are different (at a higher level) than the similarly-named "Graduate Certificate/Diploma" qualifications.
Both Certificates and Diplomas can be separate qualifications that you enrol in deliberately, commonly at polytechnics and non-degree-granting institutions. They're also often exit points for people who leave part-way through a longer qualification. These will require an undergraduate degree as prerequisite.
A Master's Degree by coursework (which ought to be what the form says, but "Masters course" is a plausible error) is:
at least 120 to
240 credits and is achieved through coursework consisting of courses, project work
and research in varying combinations. It may build on undergraduate study in the
same academic field, or it may build on the more generic graduate attributes of
an undergraduate degree in other fields, or in some cases on relevant professional
There is no thesis in a coursework Master's. 240 credits is two years of full-time study.
A Master's degree not further specified (in context here, with "by coursework" as a separate category) is either:
- a 90- to 120-point thesis following a postgraduate diploma or Honours degree (one year); or
- a 240-point qualification involving both coursework and a thesis, essentially the postgraduate diploma immediately followed by the thesis (two years, commonly officially "MSc part 1" and "MSc part 2").
All forms of Master's degree are defined on page 18.
A PhD is a 360-credit thesis, representing three years of full-time research (pages 20-21). It will generally require a Master's or Honours degree as prerequisite.
All of these are different qualifications that will be offered specifically by universities. You should pick the one that matches what you're planning to enrol in, which will be named directly as one of these categories.