From a previous post I wrote up a while back:

Potatoes were introduced into Europe in the 16th century as part of the Colombian exchange. Initially, they were extremely expensive so they were considered a delicacy. Some aristocrats and royals (edit: here I'm mostly referring to England and Iberia) even believed they had aphrodisiac qualities, which only increased both the price and desire of the plant among the upper classes. The masses, however, did not share the aristocracy's initial views and instead greeted the potato with suspicion and derision.

It's eventual spread to the masses, like so much of European history, is dictated by warfare. Warfare in the early modern period was often accompanied by starvation of the local populace as grain stores were used up and harvests were destroyed by armies on the march. In many ways, the Thirty Years War was the climax of this cycle of warfare and deprivation in Europe in the early modern period, especially in German speaking Central Europe. It is from this point that the spread of the potato is particularly well documented. It was first widely adopted in Alsace in the second half of 17th century; in Flanders at the end of the 17th century, and in Prussia during the middle of the 18th century. During the early part of the War of the Austrian Succession, Frederick II's government distributed potato seed free of charge to all parts of his realms and greatly encouraged cultivation of the crop. When French, Austrian, and Russian armies invaded Prussia a short time later during the Seven Years War, it was in large part thanks to the potato that Frederick's peasants were able to weather the storm. The invading armies took note of this new crop and brought it back with them, and those countries began, more or less, encouraging their own farmers to plant the crop. In German speaking Central Europe, the potato was again proved to be so good at keeping the masses fed that the War of the Bavarian Succession (1778-9) was also known as the Kartoffelkrieg ("Potato War") because, although the armies (Prussia v. Austria) themselves basically stalemated, everyone went home well fed, thanks to the starchy root vegetable. As a result, primarily of Frederick's exploits, the potato went from either unknown or despised by the masses in much of Europe at the beginning of the 18th century to a major part of the European diet by the end of the century.

Its spread in France was a bit slower than elsewhere, however, but it is also related to war. Antoine-Augustin Parmentier was taken prisoner by the Prussians during the Seven Years War and survived for three years in captivity on a steady diet of potatoes. When he was released in 1763, he was not much worse for wear either. So he returned to France determined to introduce the potato to France. He began with a number of scientific studies proving the nutritional value of the crop. When harvests failed in 1770, the Besancon Academy offered a prize for the best study of food substances that could help alleviate the calamities of famine, Parmentier entered his own potato study and won the prize. In 1775, after the so-called Flour War, Parmentier began trying to get government backing for use of the potato to help with the food shortages in Paris. It wasn't until 1785 that he got that royal seal of approval. So Parmentier began by wooing the high society. He hosted dinner parties (guests included Ben Franklin and Thomas Jefferson) and even managed to get the potato served at court. But given the popularity of the aristocracy in Ancien Regime France, this did little to persuade the Parisian masses. His royal seal of approval, however, permitted him to plant a whopping 40 acres (which yielded literally tons of potatoes). And he did do a bit of trickery to get the common folks interested. He posted guards and had the guards shoo away anyone coming near the fields, but then left the fields unguarded at night. Obviously, inquiring minds returned at night to see what was so valuable and many absconded with the potatoes. Over the next decade or so, eating the previously despised crop became routine, such that Parmentier was inducted into the Legion d'Honneur as one of its first members in 1802, primarily as a result of his work with the potato.

Source: Reader, John. Potato: A History of the Propitious Esculent. New Haven: Yale University Press, 2009.

I can only really speak about the Irish experience here, but I understand it's reasonably representative of the overall European process.

You are entirely correct that people were slow to adopt the potato. Indeed, it was grown for its flowers before it was grown for food, and the flowers appear in some European heraldry. Part of this was because the tubers were thought ugly an unwholesome, and part because the plant (like the tomato) looked like poisonous nightshades. It was grown for animal fodder for a short period, around 1590-1600. After this, however, it was discovered that the potato will grow almost anywhere in Europe.

This was the principal and major reason for its adoption: it really will grow in places that nothing else edible will. You can plant it in bogs, on mountaintops, terraced into slopes where nothing but a goat will otherwise go, in barrels, in buckets, on a roof. And so on. In the tight confines of Ireland, where a great deal of land is unsuitable for any other crop, it made a vast difference. This was true across Northern Europe as well. Further, it keeps really well, so storage for over-wintering was genuinely a matter of heaping them up in the dark.

The second reason is that unlike almost any other carbohydrate, it requires next to no preparation. You pull the potato out of the ground, you wash the soil off, you boil it or bake it, and you eat it. Compared to the shenanigans you need to engage in to turn, say, wheat into something you can eat, it's a miracle food.

And the third reason is that until the coming of the blight - which was, astonishingly, probably from one single tuber imported from Mexico via the US in the late 1830s or early 1840s - it was a very disease resistant plant, most likely because there was nothing in Europe that was inclined to attack it.

So for 200 years or so, you got some potatoes, you stuck them in the ground, they multiplied, and you repeated, feeding your family from land unsuitable for anything else. Among other things, this meant that whole swathes of Europe were covered in very specific varieties of potato, like the Irish White Lumper. This meant that when the blight did come, there wasn't even a genetic variation to save the crops - they fell like dominoes.

Sources:

Cormac Ó Gráda, The Great Irish Famine, Cambridge University Press, 1995.

Redcliffe N. Salaman, William Glynn Burton, The History and Social Influence of the Potato, Cambridge University Press, 1985.

Vida Carmen Kenk, 'The importance of plants in heraldry', Economic Botany July 1963, Volume 17, Issue 3, pp 169-179.

S B Goodwin, B A Cohen, and W E Fry, 'Panglobal distribution of a single clonal lineage of the Irish potato famine fungus', Proceedings of the National Academy of Sciences of the United States of America, November 1994, Volume 91, No. 24.