Succession is the process where the title of King or Queen passes from one person to another. The line of succession is determined by the order of birth of members of the Royal Family.
The Royal Succession Act 2013 outlines the law of succession in New Zealand. This is almost identical to the law of succession in the United Kingdom and other Commonwealth Realms.
Charles, the former Prince of Wales, was first in line to the Throne after his mother, the late Queen Elizabeth II. That is why he has become King in her place.
The King has granted Prince William and Catherine the titles Prince and Princess of Wales. William has also succeeded his father as the Duke of Cornwall and the Duke of Rothesay in Scotland, so the Prince and Princess of Wales are also now officially known as the Duke and Duchess of Cornwall and Cambridge.
You can find the current line of succession via the following link: Succession | The Royal Family