To one extent or another, corals do require a certain amount of light in order to survive. Some corals, however, depend less on light than they do nutrients extracted from the water column for their nutrition.
Most soft corals, zoanthids, 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 and target feed them with a turkey baster.
Many corals will benefit from the food that you feed the fish and invertebrates in your tank. When meaty foods float by or land on corals, they will be consumed if the food is desired by the coral. Copepods, Amphipods, Brine Shrimp and Mysis Shrimp will also be consumed by many corals. Copepods and Amphipods are actually quite easy to cultivate in refugiums. Brine Shrimp eggs can be inexpensively hatched and grown in a simple DIY Brine Shrimp Hatchery. It is difficult to generalize the food requirements for groups (LPS, SPS or soft) of corals as there are always a few renegades in each group which have a more selective diet. We highly recommend obtaining a good book (reference source) on corals to determine what your specific corals feed on. One book which we highly recommend is "Aquarium Corals - Selection, Husbandry and Natural History" by Eric H. Borneman. The sections devoted to each coral provide detailed information on what the corals feed on in the wild. There is also an excellent chapter devoted to feeding corals in your aquarium.