New Zealand Birth Records
Genealogists should be glad that European births have been registered in New Zealand since 1848, not too many years after its original white settlement, since this means that the vast majority of births in the country are easy to trace - a relief for anyone piecing together a family history.
For anyone looking on the Maori side though, things are tougher; it wasn't until 1913 that New Zealand required the registration of Maori births (although it's certainly possible to find a few records of births before this time), and in the early years enforcement was lax, which meant that many Maori births actually went unregistered.
The Registrar General's IndexYou can find all European births after 1848 in the Registrar General's Index of Births. However, remember that births in the index are listed by the year of registration; that might not necessarily be the year of birth, especially say, for babies born in December.
For the years 1848-1854 there's just a single index, an indication of the country's small population and number of births. As New Zealand grew though, from 1855 until 1956 each letter of the alphabet was given its own page for each year. The folio numbers used in the index indicate both where the birth was registered and in which quarter of the year (since 1956 the index has included a column showing the place of birth). Unfortunately, the indexes aren't accessible online (you can request an index search, although you'll have to pay for it), but some can be found in larger public libraries in New Zealand. For complete indexes, you'll need to go to the office in Wellington.
When you've established the entries you want, contact Births, Marriages and Deaths for a certified birth certificate or a printout.
Information On Birth CertificatesThe birth certificate gives you plenty of information, not only about the baby, but also their parents. Certificates issued before 1875 list the birth date and place, the child's sex and name, the first name and surname of both the father and mother (plus the mother's maiden name), as well as the father's rank and profession. The certificate will also contain details of the birth informant and when and where the birth was registered. That's enough to make any family historian gleeful and push research further, but after 1875 things became even better, since birth certificates were amended to include the age and birthplace of both parents - particularly useful in tracing immigrant parents. From 1912, certificates also listed the previous children from the parents' marriage, both living and dead. This offers you a very full picture of the family at the time to flesh out the family history and clues about children who might have died young, which can be especially helpful in a time when infant mortality was high.
Other SourcesWhere you can find a good one, old baptismal registers can be useful. Often they're still at the church, or you might find them in regional records centres, but in many cases the Registrar General has copies. The problem though, is that whilst some contain plenty of information, others are sketchy at best - there's absolutely no consistency. You should also remember that they'll contain the date of baptism, and not necessarily the date of birth.
You might also try local newspapers from the period, many of which will contain birth notices.