At least in the US, and probably in Europe, "nightshades" means "tomatoes, potatoes, peppers, and eggplants." I'll break these down and go over (roughly) what they do and don't include. One reason for this listing is simply species diversity in our own food crops (to make things even harder, there are plenty of nightshades grown as food crops in other parts of the world that don't fit in one of these four categories). A second reason is that sometimes we use words unhelpfully. "Pepper", for instance, is not always in the name of everything we would agree is a "pepper" (e.g. pimientos), but on the other hand is in the name of some things that aren't nightshades (e.g. regular black pepper).
I will also give an extremely brief description of what to watch out for, since nightshades are frequently used as flavorings or ingredients, so just knowing what tomato looks like, for instance, won't help you much. This is true for lots of other common allergies (milk, eggs, wheat, nuts), too. I expect most posts in this group will be devoted to fleshing out this information.
Examples: tomatoes, tomatillos
Watch out for: sauces, soup broths, things made with soup broths, powdered flavorings on snack foods, red things, marinated things, things cut with the same knife, foods with lots of ingredients
Examples: white, new, and blue potatoes
Does not include: sweet potatoes, yucca, jicama, parsnips, anything in the genus Batata (several hundred varieties of south american root vegetables), true yams (the 6-foot long member of the lily family, not the orange things you eat for thanksgiving), radishes, squash, turnips, rutabagas
Watch out for: soups & stews, snack foods, potato flour in breads & breadings (this is a major problem), food marketed as "gluten free", modified food starch
Examples: bell peppers, chilis, pimientos, paprika (paprika is a particular kind of dried bell pepper powder, not a separate plant)
Does not include (this gets ugly rapidly): Black pepper, white pepper, at least some things labeled "green pepper", at least some things labeled "red pepper" (most crushed red pepper is a chili pepper, but it is a mystery to me whether the red pepper sold in whole "peppercorns" is ever, sometimes, or always the kind from Piper nigrum), "pink peppercorns" from plants of the genus Schinus, and really a bunch of other things with "pepper" in their names
Watch out for: anything spicy, anything red, anything with flecks of red, marinated things, sausages
Watch out for: baba ganoush, specialty hummus, food marketed to vegetarians, fried things in Chinese restaurants that you think are turnip but aren't
Note: There are at least two species of cultivated "eggplants". Our familiar variety of Solanum melongena is big and purple, and some varieties of this purple kind are sometimes known as aubergines. A second species also called "eggplant", Solanum gilo, is a Brazilian eggplant, also called scarlet eggplant, gilo, or jiló. Brazilian eggplants look a lot like the Thai variety of our familiar species.
Selected other miscellaneous nightshades
General note: There are a lot of these that Americans will probably never see in person. I am adding this section to show a) the world is big, b) there is lot to know beyond what I have written here, and c) this advice breaks down and is not so useful if you travel to South America or Asia.
- I have never seen any of these for sale anywhere:
- Melon pear, AKA pepino, pepino melon, or tree melon. Primarily from Peru and Chile. Some name confusion: papayas, which are not nightshades, are also called tree melons, but fortunately they look very different. On the other hand, "pepino" is the common name for at least two species of nightshade; I have no idea if the others are edible by non-allergic humans. Sort of round and light green.
- Naranjillo or lulo. Primarily from Ecuador and Colombia. Looks like a tomato.
- Other misc tomato-like species not available commercially: wild tomatillos, currant tomatoes, tree tomatoes, strawberry tomatoes
- Other misc potato-like species not available commercially: wild potato, Colorado wild potato
- A ridiculous naming problem: The "garden huckleberry" is a nightshade, but the regular kinds that you are personally likely to grow in your garden are all species from two different genera of heath, not nightshade. All of these things get used in preserves, jams, and pies, but the nightshade kind is unlikely to turn up here in the US.
You can learn more from the Plants for a Future database and the Wikipedia page on the genus Solanum. These are fun and I encourage everyone to read them, but don't go in thinking they will definitely help you not die or kill anyone.
Miscellaneous things that aren't at all like nightshades but that people often get confused about (very incomplete list)
These are not nightshades!!!
Onions, shallots, leeks, garlic (lilies, all), horseradish, wasabi, ginger