The Potato (Solanum tuberosum) is a perennial herbaceous eudicot cultivar of the nightshade family (yes, potatoes are actually related to freaking nightshade) native to the Andes. Potato plants usually grow about 60cm tall and produces small toxic fruit which bear a vague resemblance to the Tomato plant (a close relative to the Potato). First domesticated around 7,000 to 10,000 years ago Potato plants are cultivated for their starchy tubers, which an individual plant can produce several dozen which contain most of the nutrients that humans require to survive and is a highly productive crop per square kilometer of arable land in temperate latitudes. Potatoes can be grown from seeds or by clone propagation by planting a tuber or sections of tubers and each plant can be harvested multiple times by replanting the plant. The Potato is a common staple foodstuff in many regions around the world prepared in a wide variety of ways, you can boil 'em mash 'em stick 'em in a stew.
Hold the Phone!?
You are probably thinking something to the effect of "I'm here reading about wizards and knights and swords and dragons and stuff and out of the blue comes a page about root vegetables? Why the hell is up with this?" Well, like many things here, it comes down to pedantry.
People often think of potatoes like bowls, hammers and brooms: things that have been around forever and don't need much explanation. They've always been here as long as people have been farming. Many fantasy writers have served knights baked potatoes at their inns as a background point and even Tolkien had Samwise expounding on the simple joys of a plate of chips. This is contrasted with things like firearms which Everyone Knows is a comparatively recent inventions and the presence of which is seen as a separating point between the Days of Old and the Early Modern World. The rub about Potatoes was that for most people they showed up well after guns did. As mentioned above, Potatoes are native to the Andes region of South America and did not really spread from there as that region was in the tropics but fairly cool despite that due to it's altitude, leaving aside the fact that it separated from Eurasia by thousands of kilometers of ocean. The first Europeans that ever saw a Potato was Pizarro's gun armed party of Conquistadors in the 1530s. The first potatoes to reach Eurasia were surplus rations on board Spanish treasure ships that were sold off and and cultivated by Spanish peasants and grown ornamentally.
A lot of history is framed as the clash of civilizations and of great men doing great things, though more low key things that people don't pay much attention to working in the background can easily eclipse these guys. It took a while for the Potato to catch on since the plant's stem and fruit are both toxic and plain potatoes are rather bland, but when it did it was a game changer for European agriculture. It grew well in places where wheat did not, it could be grown in wheat fields when were left fallow, land which was not well suited for wheat or barley and was less susceptible to sudden frosts or armies burning fields (a rather common tactic in modern warfare and one which racked up a very nasty kill count as well). Once extracted it was ready for cooking and they keep well if stored in a cold, dark place. This caused a spike in Europe's population and laid the foundations for the Industrial Revolution.
In short, potatoes (alongside corn, tobacco, tomatoes, green beans, pumpkins, chocolate, strawberries, rubber, vanilla and a host of other things) did not reach Europe in medieval times. Strictly speaking this is not a problem with most fantasy worlds as it's history is not a one for one parallel with real world history and there is no reason why you can't have Brave Chivalrous Knights, Beautiful Elven mages, Proud Dwarven Guardians and Mighty Orcish Warriors with swords, axes, bows and staff who kick ass, complete quests and celebrate their victories with a basket of fries. It probably means that the population density is a bit higher than the actual middle ages, but this is generally not something to get worked up about in the majority of situations that would be pertinent for most games. If you want for a bit more authenticity you are free to omit them from your game world as well. Which does not stop there are some people from making a stink about it, either in terms of "Why do you have New World cultivars in a pre-Columbian world you uneducated peon!" as well as "Fuck you for saying that any game should ever be concerned with authenticity or fine details!" in response. Of course, your fantasy world may not have continents based on Eurasia and the Americas, or it may do but they have been trading for considerably longer than in Earth history. Also, a generally feudal lifestyle continued in Europe until as recently as the first world war, so many rural populations did not interact much with industrialisation, especially in areas far from international ports. Thus, even if your fantasy peasants do grow Earth crops (to save you inventing multiple whole crop families and histories) there is no reason to assign those crops the same origins as they had in real history.
As A Meme
- Potato Song: In the Lord Of The Rings: The Two Towers movie Samwise Gamgee explains to Gollum how you can “boil ‘em, mash ‘em, stick ‘em in a stew” after claiming all he needs is a “few good taters” while Gollum rejects his idea in favor of eating fish raw. This marks the first known cinematic depiction of the cultural conflict between the River Hobbits and the more common Hobbits. This exchange was the inspiration for a famous techno remix on YTMND. Watch it here.
- Death Of Fantasy: In the context of humor, potatoes are apparently the difference between fantasy and fantastic science fiction. This meme persisted on /tg/ for some time, with spoiler tags often being used when not simply the OP idea of the thread in the first place. Whether as magical relics, the creations of a mad wizard, a Pandora’s Box from bored gods, or something Eldritch, the potato discovery is of great importance. While posted constantly in its heyday in a way similar to "Excuse Me" Guy, it never earned the same amount of Skub and had a far shorter lifespan. The image accompanying these threads was usually from the ending of the anime Spice And Wolf where a merchant and his goddess partner obtaining a potato in a box heralds the end of gods and the rise of man.
- Potato Battery: If your computer is underpowered or if your aim is shitty. It's referred to being "potato". This variation comes from the fact, with a small enough light bulb and a large enough potato. One could use the potato to power a bulb.
- During the wars of butthurt between Warhammer Fantasy and Age of Sigmar, or at least mostly the trolls who took advantage of tension between people who were mutually exclusive fans of them, a term popped up, "Potato Sack". The general idea is that the models of Warhammer Fantasy look like the stereotype of those living in poverty in the 20th century who would use scraps of burlap, easily obtained from bags that things like potatoes or scrap metal would be sold in, for clothing. It refers to the more ground-level peasant common troops seen in almost every army of Warhammer Fantasy (Elves don't count) compared to the armies made exclusively of walking demigods being far more common in Age Of Sigmar. Used in reference to either, it aggrandizes Age Of Sigmar design while degrading the Warhammer Fantasy design. Thanks to overuse Warhammer Fantasy fans quickly began to use it as a sign of pride, rendering it a dead meme (as far as its original intent). It is worth noting that burlap wouldn't have existed in the era that Warhammer Fantasy is supposed to represent, although AoS is a bit too Spelljammer to know for sure it doesn't exist there.