Tectonic forces (plate tectonics) move plates of Earth's crust around slowly but constantly. Occasionally all of the continents are shoved together forming a supercontinent. Earth scientists conclude that this has happened a handful of times throughout history, and the most recent supercontinent was called Pangea. It formed around 300 million years ago which geologically speaking was only just before the appearance of dinosaurs. The map below shows the relative positions of the modern continents within Pangea.
("Pangaea continents" image courtesy of Kieff - File:Pangaea continents.png. Licensed under CC BY-SA 3.0 via Wikimedia Commons - https://commons.wikimedia.org/wiki/File:Pangaea_continents.svg#/media/File:Pangaea_continents.svg)
Pangea persisted as a supercontinent until about 200 million years ago when tectonic forces caused the continental plates to be shoved around and apart and moved them to their current locations.
There are many maps available that show where the different continents were at different times before, during, and after Pangea, but geoscientists recently released an extremely cool computer animation that shows the break-up of Pangea in one million year time increments. You can run this animation by clicking the link below which will take you to the AAAS Science web site where there is also a short, readable article that explains what the animation shows.
Animation of the break-up of Pangea
Short readable article about the animation and the break-up of Pangea
My personal favorite part of the animation is at the very end where the Indian sub-continent collides with southern Asia - this is what formed the Himalayas...a mountain range that is still building.