New Zealand study Visa Guide & Assessment:

New Zealand offers a huge margin of studying at internationally acclaimed and recognized institutes. Taking admission in a school and undergoing tertiary studies in New Zealand can make you eligible to receive a student visa. If you are an Australian permanent resident or reside permanently in New Zealand then you do not need a student visa to study in New Zealand.
Being in possession of a visit visa which is either 24 months can also make it possible to study without a student visa. But the duration of the course should not be longer than 3 months during one calendar year.
Studying in New Zealand for three or less than three months is possible with a visit visa, but if you intend to study for more than three months, then you should be in possession of a student visa. But even if you are enrolling in a short course which is less than three months long, then that course has to be either approved or exempted by the New Zealand Qualification Authority.

New Zealand also has specific exchange student schemes. If, you are offered to be a part of such a scheme, then you can eligibly be granted a student visa for the time period that the scheme is in effect. But you need to have been approved in the program prior to applying for the visa.
Children on a visit visa are also allowed to attend school without student visa, but they cannot attend for more than 3 months in a period of 12 months.
If you are a part of the Working Holiday Scheme then you are allowed to take admission in a course which is of duration no more than 3 or 6 months. You do not need a student visa for this.

In order to apply for a student visa you need to be enrolled in a study program offered by the NZQA or the New Zealand Qualifications Authority. Other necessary conditions; like fee payment and documented proof of necessary arrangements needs to be provided while applying.