There are two types of crinoid; stalked or sea-lilies (Photo) and un-stalked or feather stars (Photo). Stalked-crinoids attach to a substrate and generally stay there (they have been noted to crawl across the sea floor), but our friends the un-stalked crinoids swim beautifully as we see in this gif. During their larval stage, they attach for longer periods of time than when adults.
Crinoids, as mentioned above, are a group of Echinoderms, and so related to sea stars, sea urchins, sand dollars, and sea cucumbers. They like to hang out on coral or sponges and grab plankton in the water column as they float or swim by. If the area they are in isn't doing it for them anymore, they take off! As in OP's beautiful gif.
They are mostly found in shallow-waters, but there is a species found at depths of 650 ft. Scienceing.com has a great write up here.