Prior to the invention of the steam engine, most products were made by hand. Water wheels and draft animals provided the only "industrial" power available, which clearly had its limits. The Industrial Revolution, which is perhaps the greatest change over the shortest period of time in the history of civilization, was carried forward by the steam engine.
The concept of using steam to power machines had been around for thousands of years, but Thomas Newcomen's creation in 1712 was the first to harness that power for useful work (pumping water out of mines, for the most part). In 1769, James Watt modified a Newcomen engine by adding a separate condenser, which vastly increased the steam engine's power and made it a far more practical way to do work. He also developed a way for the engine to produce rotary motion, which may be just as important as the efficiency gains. Thus, Watt is often considered the inventor of the steam engine.
Newcomen's and Watt's engines actually used the vacuum of condensing steam to drive the pistons, not the pressure of steam expansion. This made the engines bulky. It was the high-pressure steam engine developed by Richard Trevithick and others that allowed for steam engines small enough to power a train. Not only did steam engines power factories that made the rapid production of goods possible, they powered the trains and steamships that carried those goods across the globe.
While the steam engine has been eclipsed by electric and internal combustion engines in the areas of transport and factory power, they're still incredibly important. Most power plants in the world actually generate electricity using steam turbines, whether the steam is heated by burning coal, natural gas or a nuclear reactor.