Great question, the answer to which relates to the various processes by which mountain chains are formed.

As you probably know, continents are components of tectonic plates, the seven major pieces making up Earth's thin, rigid outer rind (called the lithosphere). Tectonic plates have moved slowly but steadily over geologic time, usually a few centimeters per year, carrying along the continents within them. (This causes the continents to "drift," giving rise to the expression "continental drift"--a better name for which is plate tectonics.)

Tectonic plates, including their continents, move with respect to one another in one of three ways: slide past one another, pull apart from one another, or collide. When two continents collide, they crumple along their edges in the collision zone, creating a mountain belt. (Continental crust--the topmost layer of the lithosphere--is less dense than oceanic crust. Consequently, continental crust floats higher on Earth's surface than oceanic crust, generally projecting above sea level. An analogy would be having equivalent sized blocks of wood and styrofoam. If you threw these blocks into a pool, the top of the less dense block--the styrofoam--would ride higher above the surface of the water than that of the denser block, the wood.) The edges of the two colliding continents crumple because both are equally buoyant, which keeps one from being pushed beneath the other. The most famous example of an ongoing continental collision is the one between India and Asia, which is producing the Himalaya--a very young, and still growing mountain belt.

Finally, to get to your question, the Urals are the result of an ancient continent on continent collision, one between the eastern edge of the supercontinent Laurasia (essentially what is today North America and Europe) with the former continent of Kazakhstania, 250-300 million years ago.

Of course there are other mechanisms by which mountain ranges are formed, but we'll leave that for another day.

Your curiosity is greatly appreciated!