The continents have been in constant motion for billions of years. They are currently drifting apart, and have been doing so for at least the past 200 million years. The driving force behind this movement is the convection currents in the Earth's mantle.
The convection currents are caused by the heat from the Earth's core. The heat rises to the surface and cools, causing the material in the mantle to sink back down. The movement of the mantle material causes the tectonic plates to move.
The tectonic plates are the pieces of the Earth's crust that move around. The continents are located on the tectonic plates. As the tectonic plates move, the continents move with them.
There are three main types of plate boundaries: convergent boundaries, divergent boundaries, and transform boundaries.
Convergent boundaries are where two tectonic plates collide with each other. When two continental plates collide, they can form mountain ranges. When an oceanic plate collides with a continental plate, the oceanic plate can be subducted under the continental plate, which can cause volcanoes and earthquakes.
Divergent boundaries are where two tectonic plates are moving away from each other. When two oceanic plates diverge, new oceanic crust is created. When a continental plate diverges, a rift valley is formed.
Transform boundaries are where two tectonic plates slide past each other. Transform boundaries can cause earthquakes.
The movement of the continents has had a profound impact on the Earth's climate and on the evolution of life. The collision of the Indian and Eurasian plates, for example, created the Himalayas, which have a major influence on the Asian monsoon system. The opening of the Atlantic Ocean, on the other hand, allowed for the exchange of plants and animals between the Americas and Europe, which led to the evolution of new species.
The continents will continue to move for billions of years to come. The current configuration of the continents is just a temporary snapshot in a very long process.