The continents are defined by convention rather than by definition, so different geographers will employ different criteria to explain what separates a continent from any other landmass. The most common definition is that the area has to be large enough that it has its own geographical features that it doesn't share with neighbouring landmasses... For example, a particular mountain range or a specific river would have to belong to one continent and not another. Sometimes, we determine which continent a particular landmass belongs to strictly by which tectonic plate that landmass is situated upon as generally the landmasses that we refer to as "continents" tend to occupy their own tectonic plate, but both systems have flaws and therefore exceptions.
In the Western World, we generally recognise seven continents: Africa, Antarctica, Asia, Australia, Europe, North America and South America. But even a person with rudimentary geographical knowledge can see that Europe and Asia are part of the same landmass. And prior to the excavation of the Suez Canal and the Panama Canal, Africa was conjoined with Eurasia and North and South America were adjoined and formed a single landmass as well.
So, in the strictest sense, it could be argued that there are really only 4 continents: Afro-Eurasia, The Americas, Antarctica, and Australia.
But geography and history always go hand in hand, and historically, mainland Africa has always been very different from Eurasia - not only is the geography different, but in terms of religion and culture, language, politics, essentially in almost any of the fundamental ways that you could demonstrate dissimilarities between two places and general differences between people, mainland Africa and Eurasia are different.
But Europe is also different from the rest of Asia. Historically, politically, in terms of religion and language, Europe and Asia don't have a whole lot in common. So despite being part of the greater Eurasian landmass, Europe is almost always considered a separate continent.
Geologically, Australia is believed to be the oldest of all the continents. Some of the rocks found and studied there have been dated and are proven to be older than those found anywhere else on Earth. That leads scientists to assume that Australia was around longer than any of the other continents, but that doesn't mean that theory won't be disproved tomorrow.
What we do know is that Australia is the smallest (by all definitions), hottest (on average), and driest (on average), of all the continents, and that in terms of the human geography, the people - the indigenous people anyway, were very different from people anywhere else in terms of their culture and language. That alone merits Australia being considered its own continent. Evidence suggests that Australian Aborigines reached Australia around 40,000 years ago. And many of the different societies spread out across the great expanse of Australia were living traditional lives, the way they had been for tens of thousands of years, when Europeans arrived.
Scientists and historians who adhere to the theory that the New World was populated by people who migrated from the Old World across the Bering Sea hypothesise that it took fewer than 200 years for human beings to reach the southernmost tip of South America, and there are some that argue that the feat was accomplished in half that time. But there is also evidence to suggest that parts of North America were inhabited thousands of years before the Bering Strait crossing came into existence. If that's true, then it lends credence to the theory that the Americas were populated before human beings - or their ancestors, arrived from Eurasia. It's also not completely out of the question to suggest that the first people to inhabit the New World might have come from Africa, not mainland Eurasia, though there is no evidence to support that theory at this time.
Australia is home to many species of flora and fauna that don't exist elsewhere. That's a clear indication that life developed differently there than in other places.
Australia is often considered an island because it's so much smaller than the other continents in terms of area.
That explains why we tend to see Australia as a continent and an island at the same time, which is why it's often referred to as "The Island Continent."
There are other sizable landmasses out there, notably the island of New Guinea, Australia's closest neighbour, and a place that's also very unique in terms of flora and fauna and human history and geography, so New Guinea is often referred to as a "Mini-Continent."
There's also the Indian Subcontinent which is a peninsula and separated from the rest of mainland Eurasia by some of the highest and most ruggedly impassable mountains we have on this planet. That means that the Indian Subcontinent bears many of the hallmarks of a proper continent, but is always considered part of Asia because it shares a land border with Eurasia.
The long and the short of it is that we have continents by definition of convention, not by any strict geographical, geological or scientific definition.