Minestrone is a thick soup made with a broth and vegetables and at least one vegetable that adds body, like beans or potatoes. The “one” ending (pronounced “onay”, remember to pronounce the ending vowels in Italian!) indicates “big”.
The waitperson will usually bring some olive oil to be drizzled on top with the minestrone.
While it is interesting to listen to folks debate the “real” recipe for minestrone, in fact making it is more of a process. It uses available seasonal ingredients. So, no, it doesn’t have to have pasta or potatoes in it, even if your mom made it that way. Each cook makes a different version at different times, so the variations are pretty endless.
The word derives from Minestre, which has come to mean a sometimes chunky, substantial soup. The big one, minestrone, is usually thicker.