Here is the short map. Our full guide to Vietnamese rice varieties goes much deeper, but this is enough to shop and cook sensibly.
Everyday and fragrant rice. Long, slim grains that cook up separate and soft with a gentle aroma. This is the rice for the family bowl, for fried rice the next day, and for porridge. Names on bags vary a lot, and the word "jasmine" is used loosely, so look at the grain and smell the bag rather than trusting the label alone.
Glutinous (sticky) rice, gạo nếp. Short, plump, opaque grains that fuse when steamed. Despite the name there is no gluten in it; "glutinous" refers to the sticky texture. It becomes xôi, the filling in bánh chưng and countless sweets. It wants soaking and steaming. See the sticky rice and xôi recipe.
Broken rice, gạo tấm. Not a separate variety but grains that broke during milling. They cook faster and soak up flavour, which suits cơm tấm.
ST25. A fragrant variety bred by Hồ Quang Cua and his team in Sóc Trăng, in the Mekong Delta. In 2019 it won the title of World's Best Rice at a rice conference in Manila. It is a fine rice, and also a name that gets stuck on bags it does not belong on, so buy from a source you trust.