Coral is an animal, yes. Some coral eat zooplankton, others live off the algae in their cells. Some coral are stationary - they don't move. Others move, but very slowly, compared to, say, piranha. Many corals will throw out 'stingers', like jellyfish, to harm other corals coming too close to their 'space'. Other corals have defenses to keep fish from eating them - poisonous tentacles, for one. You've heard of the sea anemone? Stings everybody except clownfish - and that's only because the clownfish has a protective coating against the stinging cells. Clown gets a nice home and the anemone gets food from the clown.

As for its status as an animal in the overall order, I don't know. It's definitely more like a jellyfish than a salmon, but that's about as much as I know.