The answer seems to be: it depends.
Down can be collected in a variety of ways. Birds which provide the feathers may be used for other purposes, for example to provide meat. Some birds are killed solely for their down, while some birds (particularly some geese) are periodically live-plucked of their breast feathers. Some birds, such as the eider duck, line their nests with down, and such down is harvested safely after the young leave the nest.
I always assumed down came from geese, but apparently, down from other types of birds is used as well.