The answer is simple: as many as we want to have. The CGPGrey video nails it, but here is my take.

North and South America are two separate continents. They have different flora and fauna, are barely connected and it makes sense in the geographic sense to make them separate.

Africa is its own space with its own geography and biology so it makes sense to be on its own.

Antarctica, with the ice sheet, is a large landmass on its own continental plate with its own geography and biology. So it gets to be its own continent.

Australia, including New Zealand and everything below the Wallace Line is its own continent.

So there are the five easy ones. Now to Europe and Asia. They are not a separate continental plate, the biology on the steppes in the center is all its own and spreads from China to Hungary, and if it was anywhere else we would call it one landmass. But there is a case to be made to segregate Europe from Asia and that is culture. The forests and rivers and mountains of what we call Europe developed a different people than the steppes, so should it be its own place? If so, why is India not a continent by the same measure (different culture, geography, on its own plate, completely different biology from 'Asia')?

So yes, there are 8 continents. India, Asia and Europe make the last three.

Why is Greenland not its own continent then? Greenland has the same geography as Northern Canada, the same biology and the same climate. And it sits on the North American Plate. Also Greenland is big, but less than 1/2 the size of Australia. So no, it is not a continent.