In countries that use a Parliamentary system, such as New Zealand, the senior position of the Executive branch of government is split into two separate roles:
1. Head of the Government. This is the one who actively runs the country on a day to day basis. He or she will command a majority in Parliament (or have a working coalition with other parties), as a government cannot be formed or properly function without one. The term of office is usually a modest period, to more accurately reflect changing views among the populace. The office is intended to reflect the will of the people through a particular leader of a political party.
2. Head of State. This is the one who is considered to represent the nation as a whole, while still having important duties to discharge (such as diplomatic conferences). It is a less powerful and more ceremonial position, which generally has less partisan stigma attached to it. If there is a vote of no confidence in the administration, the Head of State can often dissolve the government and call new elections. The term of office is usually a longer period, to aid in overseeing the stability of the nation. The office is intended to unify the people and facilitate peaceful succession from one government to the next.
In New Zealand, our Head of Government is the Prime Minister, and our Head of State is the Monarch of the United Kingdom (currently Queen Elizabeth II). Since the Queen can hardly ever actually be in New Zealand, a representative of the monarchy is chosen in the person of the Governor General. The Governor General however is not an adequate substitute for a New Zealand Head of State. The Governor General has little real power, having an almost purely ceremonial role. The Governor General is essentially appointed by the Prime Minister, and can also be dismissed by the PM, thus eliminating any possibility of the Governor General holding the Prime Minister accountable for his actions. There is very little that restrains the Prime Minister, aside from elections and the rare occasion of a usually toothless judicial review.
We need a republic to elect our own independent kiwi Head of State that can adequately represent New Zealanders while providing a real check on the Prime Minister's power.