"Quilt" technically refers to the construction of the item: an insulating middle layer, such as a blanket, is sewn between the cover and a backing, usually by the sewing technique known as "quilting". http://en.wikipedia.org/wiki/Quilting
Eiderdown is a type of feather that comes from eider ducks. An eiderdown comforter, or pillow, or featherbed, or whatever else, would be filled with eiderdown.
Here is a local interpretation of these terms. I suspect there is considerable variation.
A cover could be anything. A bed-cover is probably a bedspread.
A bedspread is used if you sleep under sheet & blankets. The bedspread is a decorative layer covering the blankets and hanging down on either side of the bed - probably not as far as the floor. It could be a single layer of fabric - probably quite heavy.
An eiderdown is a thicker cover and should, as the name suggests, be stuffed with feathers. It may cover the top of the bed only.
If you don't sleep under blankets, you sleep under a duvet.
A duvet is thick, soft and full of something like feathers or synthetic insulating stuff. A duvet is put inside a duvet cover.
What's a quilt?
Tricky question. A quilt around here could be any of the above, though probably it is a somewhat padded form of bedspread with decorative stitching or indeed made of patchwork. MrsP went through a fanatical patchwork phase that resulted in patchwork quilts all over the house.
Which is most common - again, personal comments.
Currently, I would say that duvets dominate and quilts come next. Bedspreads are used on beds that don't have duvets. I haven't heard eiderdowns mentioned a great deal recently. Cover?