Onions family includes all the onions, garlics, and leeks, and they all belong in the lily family. Grown undergrounds, onions' fructose sugars are broken down in long and slow cooking. The leaves of the bulbs, such as leeks, chives, and onions. Here are some of the most common ones we see in recipe books.
A.K.A Yellow Storage Onion or Yellow Globe onion. This is the second most common market onion. This is grown through the summer and harvested in the fall. It is rich in sulfur (what makes people cry), but also intense flavour. This is the most standard onion for almost all types of cooking. When you can't find the specific onion called by a recipes, this would most likely be the most suitable replacement. Some people find this onion too pungent to eat raw. Spanish Onion is the larger and sweeter versions of yellow onions.
A.K.A Spring Onions, Summer Onions, or Fresh Onions. It is actually the same yellow onion as above, but grown in sulfur poor soil. It is the most common market onion, probably because it is friendlier on the eyes. The flavours are mild and more suitable to be eaten raw in salad, burgers, and sandwiches. It can tolerate some heat. They look very similar to yellow onion, except for the light yellow greenish skins. Sweet Imperial, Arizona, granex, Maui, Oso, and Carzalia onions have similar properties as sweet onions.
A.K.A Purple Onions. Chefs choose this onion for its beautiful color for salads and garnishes. This might be shocking to people: the onions is pigmented by water-soluble anthocyanins. This is why only the surface layers of each leaf scales are red and when you cook it, the color dilutes and dulls (and potentially colors your stew or soup). Sweet red Italian onion, Italian red onion, creole onion, and red torpedo onion are in the same school. You can use green onion as a replacer in salads.
This onion variety is moister and don't keep as well as yellow onions (very prone to molds.) It has a cleaner and more tangy flavour than yellow onions. I haven't had must experience with this onion variety, but it is very widely used in hispanic cooking. White globe onion is similar to white onion.
Especially prized in France and South East Asia. It has delicate, sweet, peppery, garlic like flavours. They tastes very similar to the white part of green onion. Its small size makes portions easy to control (so you don't have to leave half of a large onion to rot in the ridge), and its finer texture allows chef to cut very fine cubes for their sauces. When cooked, it has a nice smokey flavour which goes well with meat. It might be too "spicy" to eat raw. Chefs at Dirty Apron use it interchangeably with other onions.
A.K.A Scallion, spring onions. It is fresh, peppery, and has a soft crunchy textures. It is used extensively in Asian cuisines: saute, soup, noodle, seafood, and more. In Western cuisine, it is often eaten raw in salads.
There are a lot more onions I didn't cover here....