Corals, for the most part, are carnivorous, feeding mostly on small animals (zooplankton) suspended in the water column of the ocean. Copepods, polychrates, chaetognaths and larvae are the more commonly consumed zooplankton items in a coral's diet. A vast majority (up to 85%) of this food emerges from within the reef (where it is produced) in the evening and at night.
Most soft corals, zooanthids and gorgonians depend almost exclusively on phytoplankton, (small water-borne plants or algae) for their nutritional needs as well as floating plankton, detritus and slow moving invertebrate larvae, rather than zooplankton (which can actively propel itself).
The third important source of food for corals is bacterioplankton, which consists of free-living bacteria as well as the bacteria associated with various materials in the water (mucas, dead plant material, and other particulate matter) which are commonly called detritus or reef snow. Almost all corals feed heavily on bacterioplankton. Material which includes detritus, floating eggs and other material is also known as pseudoplankton.
The fourth category of food utilized by corals is Dissolved Organic Material (DOM) which is absorbed across cell membranes directly into the coral.
Many of the corals with larger polyps (i.e. Cynarina and Catalaphyllia) are capable of capturing and eating larger food items, including the occasional small fish. Many corals (particularly Gorgonians and soft corals) may select their food based more on the size of the plankton, than its composition.
In the past, it was believed that the large polyped corals, with their more efficient tentacle formations, obtained a large portion of their nutrition from active feeding on the food that floated by, rather than from their zooxanthellae algae. It has since been discovered that many of the small polyp corals are actually more aggressive feeders than their larger cousins.
If you have live corals in your aquarium, you are probably wondering what foods your corals eat to supplement the nutrition provided by their resident zooxanthellae algae. You could just make a slurry of a variety of different foods which cover the entire spectrum (the "shotgun method" approach) and load it into your tank, allowing the corals to select what they want from the mix. The uneaten food in the mix is guaranteed to increase your nitrate levels in a short period of time. Or you can fine tune the supplement to the requirements of your specific corals.