small polyp stony (the colored sticks and branches like acroporas, montiporas, stylophoras, etc) and large polyp stony (the tentacled poofy corals like frogspawn, plate, torch, etc.)
The ones with the very small mouths (polyps) are the fussiest, living very much off light (their colors come from algae that live in their tissues: the algae photosynthesizes, produces sugars, feeds the coral, which lays down more calcium carbonate skeleton and grows more mouths, more skin, etc. They can eat very tiny things like baby cyclopeeze and pods, and take in chemicals from the water, most particularly calcium) The large polyp stony operate somewhat like that, but are water-filled, with larger mouths, and some, like plate, may take in a whole cross-section of a shrimp at one gulp. They have stinging cells (nematocysts) like a jellyfish in their tentacles and over time can concentrate them in a particular set of tentacles to increase the nastiness of the sting of those tentacles, say, toward a neighbor it doesn't 'like.'