The humble potato has its origins in the Andes. Over 4,000 varieties of potato were historically cultivated in Ecuador, Bolivia, Peru, and Chile—and they have been for thousands of years. As far back as 10,000 BCE, strains of potato have been found in southern Chile. The credit for potato domestication has to go to Andean peoples whom archaeologists believe had completely domesticated the food by 2,000 BCE.
Sidebar about potato cultivation:
The potato domesticated by the early Andeans was not the same as the potato we recognize today. Potatoes can be grown in two ways, from seeds and pollinated flowers or from slices of the spud itself. Slicing a potato in half and planting each half results in two genetic clones of the original potato. Planting the seed of a potato results in new plant growth, which will cross-pollinate and give you genetically different potatoes than the parent potatoes. Most, if not all Andean potato farmers grew potatoes from seed instead of spud, which led to massive potato variety in South America.
Christopher Columbus’s trans-continental voyage marks the beginning of the Colombian Exchange, the first global exchange of people, foods, germs, ideas, everything really. One of these things was one strain of potato, S. tuberosum. The first European contact with potatoes seems to be Francisco Pizarro and co. who in 1532 saw the Indians eating small round objects, and decided to try and eat them as well. Over the remainder of the century, potato cultivation and consumption spread through Spanish hands to the Canary Islands, France, the Netherlands. The S. tuberosum strain was one of the only strains of potato that the Spanish carried over to Europe.
Potatoes stayed a relatively minor food in Europe, especially when compared to grain, until the late 1700s. 1770 saw crop failure in Eastern France. The response was a farming competition named "Plants that Could in Times of Scarcity be Substituted for Regular Food to Nourish Man". Most of the entries were potatoes, and the winner was Antoine-Agustin Parmentier, a nutritional chemist and potato activist (r/unusualcareers?). He kept pushing spuds, coincidentally at the same time that King Louis XVI lifted price controls on grain. As prices for wheat and wheat byproducts went through the roof, Parmentier was setting up potato publicity stunts. His efforts paid off and soon spuds were being served across Europe.
Widespread popularity of the potato has been credited by William H. McNeill with “feeding rapidly growing populations, permit[ing] a handful of European nations to assert dominion over most of the world between 1750 and 1950.” Potato cultivation positively correlates with population booms across Europe in the 17th-20th Centuries. Ireland’s population grew from 1.5 million in the 1600s to at least 8.5 million over the course of 200 potato-eating years.
As more of an epilogue of the farming industry than the potato itself:
Bringing over a single, cloned strain of the potato resulted in a continent-wide monoculture, making the potato blight in Europe and the potato beetle in the United States particularly devastating. The blight didn’t have to adapt at all to spread from plant to plant and field to field. The beetle found a food it liked, and always found massive fields of it. This has led to the development of large-scale genetic modification of produce and unprecedentedly large pesticide and fertilizer industries.
This answer is a condensation of “Chapter 6: The Agro-Industrial Complex” of Charles Mann’s book 1493: Uncovering the New World Columbus Created.
*Sorry for taking so long to get you a more in-depth answer, and my apologies to the mods for leaving a placeholder. Feel free to remove my prior comment, or I can delete it myself.