Conularid fossils are something of a mystery for Paleontologists because they don't seem to fit into any one particular Phylum well. Recently it's been proposed that they are a type of scyphozoan cnidarian (jelly fish). One theory has them floating in the water column and snagging passing prey while others assert a more sedentary lifestyle attached to the substrate. They are known from the Cambrian through the Triassic but I've only ever found them in Ordovician rocks north of Frankfort, Kentucky.
Here is one that my friend Mike over at Louisville Fossils sent me. It's from the Borden Formation in Jefferson County which is Mississippian in age. Note the fine ribs along the sides and the square cross section.
This specimen is from the Carboniferous (Pennsylvanian) Finis Shale near Jacksboro, Texas. Note that this fossil doesn't have the square cross section like the above specimen, rather it is more oval shaped. EDIT - This fossil is not a Conulariid but is actually a Pelecypod called Aviculopinna. I should have picked up on that myself due to the shape of the fossil but sometimes I get fooled by a label.