There are seven continents...in alphabetical order - Africa, Antarctica, Asia, Australia, Europe, North America and South America. Some people consider Australasia and/or Oceania to be continents, too. However, there is just no way these places could fit the definition of what a continent is according to The Oxford English Dictionary (TOED), which is the 'bible' of English word-meanings. It says: "One of the main continuous bodies of land on the earth's surface." It's pretty clear that, if there is one thing that just cannot be said of a myriad of separate islands, it's that they fit the description 'continuous'! The dictionary goes on to list the continents exactly as in the opening sentence above.
Australasia was a name invented by Charles de Brosses, a French cartographer, in the 1750s and Oceania was a name invented by Victor Malte-Brun, a French geographer of Danish extraction, in the 1840s. TOED does not use the word �continent' in its entry for of either of these names - Australasia and Oceania.
Significantly, nor it seems did either Frenchman imagine he was naming a continent when he created the given name, but rather just a means of identifying a body of islands. One thing you can be certain of is that 18th century scientists knew their Latin and it is perfectly clear that �continent' comes from the same Latin root as the words �contain' and �continuous' - ie �continere', literally meaning �to hold together'.
Call Australasia and Oceania �geographical regions', call them �archipelagoes', call them whatever you like...but you really can't justify calling them �continents'. (Yes, I know some people who ought to know better do, but they shouldn't!)
New Zealand is not part of any continent.