Potatoes were a crop from the Americas, so during the age of colonialism, potatoes were brought to Europe, and quickly became one of the most essential crops for a lot of countries(i.e. Ireland) why did a new world crop like this become one of the most common sources of nutrition in the old world?
The answer I'll be giving you is purely based on a functional perspective and not on historical records.
As you may know, potatoes are a great source of carbohydrates and these are the main source of energy on our diet, even though we're able to produce energy from fats and proteins too. So, despite other essential nutritional requirements (vitamins, essential amino acids, et cetera), our diet is heavily dependent on carbohydrates to supply the calories our body demands in exchange for the work it produces.
During the middle-ages, the common people diet consisted basically on what they could find near them, if they lived next to the sea or a river, they could fish; if they lived next to a forest, they could hunt; if they had domestic animal, they could eat dairy products; and so on. But bread was the center piece in their diet. I'm guessing here, but bread would probably have the best energy/calories-price ratio. Corn and wheat could be planted in most of Europe, whereas rice, for example, had more strict requirements for cultivation. And that's why pretty much every village would have the means to produce flour, be it a windmill, a watermill or hand-mills.
With the introduction of the potato, a new source of carbohydrates was available. It was relatively easy to cultivate and it didn't need to be processed. It had the slight disadvantage of being less genetically diverse, and thus more prone to disease and the resulting loss of crops. Anyway, it didn't need to compete against corn and wheat, as is could be used in crop rotation system, allowing, for example, the cultivation, in the same field, of corn in one year and potatoes in the following, without exhausting the soil and increasing resistance against diseases and pests.
Even better, potatoes can have productive crops in harsher climates (colder and more humid) and soils (harder/rockier and more acidic), in which wheat would have an hard time thriving. That would explain the importance of the potato crops in locations as Ireland (that you mentioned) and the Baltic nations, for example.
Because potatoes originate in the Andes mountains, which are very cold in the winter and warm in the summer. In other words they're perfectly adpted to the cold climate of the Northern hemisphere.
The areas where potatoes are currently most widespread are those less suitable for other crops such as wheat. That includes Ireland, northeastern Europe (Belarus, Russia) and hills all over Europe.
