Today's continents were formed when the giant landmass Pangaea separated into smaller continents, due to plate tectonics. Pangaea was a collection of earlier continents.
Initially, Pangaea broke apart into two large areas, known as Gondwana and Laurasia. Gondwana was made up of what became Australia, India, South America, Africa and Antarctica while Laurasia was composed of Europe, North America and Asia. Eventually, North America split from Laurasia, and South America split from Gondwana, pushed apart by plate tectonics. India split from Gondwana and ran into Asia, creating the Himalayas. Australia and Antarctica also broke off of Gondwana.
Plate tectonics is the theory that the Earth's top layer is composed of a series of plates that move over the hotter layers below. They are pushed apart at continental ridges, like the one in the Atlantic Ocean. Fault lines and subduction zones can also affect how they move.