Moon jellies and their cousins, which include lion's mane jellies and sea nettles, are known as true jellies. They belong to the class Scyphozoa, in the phylum Cnidaria, which also includes corals. (A phylum is such a broad taxonomic category that humans, fish, snakes, frogs, and all other animals with a backbone belong to the same one -- the chordates -- as do salps, which are sometimes lumped with jellies.) As adults, true jellies are shaped like upside-down saucers or billowing parachutes. They propel themselves through the water by contracting the muscles of their bells, and their tentacles are equipped with stinging cells that shoot out tiny barbed tubes to harpoon floating prey. To reel it into their mouths, they use streamer-like appendages known as oral arms. In some species the oral arms have mouths of their own.
Jellyfish like the dreaded Portuguese man-of-war are also related to corals, but they're part of a different subgroup, the siphonophores, which practice an unusual form of collective living. What looks like a single man-of-war is technically a colony that developed from the same embryo. Instead of simply growing larger, the embryo sprouts new "bodies," which take on different functions. Some develop into tentacles, for example; others become reproductive organs.