I suppose in general, a flight cage is one big enough for a bird to fly in a little - bigger than a normal cage, smaller than an aviary.
However, it's most commonly used to refer to a 32x21" cage that is is marketed as a flight cage. Several different manufacturers all make "flight cages" of virtually identical dimensions. The dimensions seem arbitrary but perhaps chosen to reduce shipping costs. In any case, they are popular because they are a good size for most smaller species, while still being easy to fit into the home, ans have a nice simple design making them practical, easy to clean, and cheap to ship. Basically just a nice quality, practical cage that you can usually get at a very reasonable price (in the US anyway). However, despite the name. these cages aren't really true 'flight cage' for anything bigger than about budgie size. They are great cages for many birds that are bigger than budgies, like cockatiels and smaller conures, they just won't really have room to fly in them. For birds that are out part of the day, that is fine though.