Sir Spankus, no road rage this time, okay?
Three things: Zooxanthellae, ciliated siphonoglyphs, and cnidocytes. Corals are amazing animals in that they act as benthic algae, sedentary filter feeders and sedentary suspension feeders; they pretty much cover all the bases. Endosymbiotic zooxanthellae are the excuse most non-feeding aquarist use to justify their husbandry. Zoox provide not only organic carbon in a variety of forms, they also provide the organic matrix that acts as a nucleus for deposition of calcium carbonate. The algae not only naturally leaky in terms of release of organic nutrients, but corals also apparently stimulate the dinoflagellate give up some of its photosynthate.
However, corals are extremely energy demanding animals, with high rates of metabolism: producing calcium carbonate buttresses against reef energies as well as "keeping clean" on the bottom of the sea (mucus is metabolically expensive in the quantities corals produce it). Thus, this may evolutionarily explain the ability to gather food in so many ways (unless you follow McConnaughey, who claims that skeletogenesis may occur to benefit the zooxanthellae).
Ciliated siphonoglyphs (openings on either side of the mouth) circulate water through the gastrovascular cavity. Since corals are way below the circulatory system clade on the evolutionary road, they use diffusion to get oxygen to their cells. The cilia bring water to the high surface area, mesentary-laden gastrovascular cavity. Thus, any dissolved organic nutrients that are present in the water are brought into the polyp and can be actively transported through the cell membranes of the endodermal cells, and then broken down to yield ATP in the presence of the O2 that is diffused. While reefs are oligotrophic when it comes to free dissolved organic nutrients, many species of fish specifically utilize living coral structure as nocternal or diurnal protection. The dissolved organic matter (along with the solid waste, but thats a different transport/feeding mechanism) released by the fish comes in contact with the
coral polyps before mineralizing/ammonifying bacteria. A problem with this may be the lack of efficient organic nutrient recycling in reef aquaria, thus, most of the organic carbon may reach the mineralizing bacteria before it is used by coral, thus building up the concentrations of DIN that are encountered.
The siphonglyphs can also bring in particulate/bacterial material to be consumed. The mesentaries that line coral gastrovascular cavities have incomplete sections that terminate in three lobes. These lobes are loaded with cnidocytes with nematocysts immobilize and phagocytosing cells begin processessing of this material. Mucus cells in the actinopharynx (inside the mouth) can also produce a mucal net or mucal sheets that is/are distributed by the siphonoglyph cilia (the polyp is usually closed). This is also likely how detritral and solid waste material is handled. This net can be withdrawn with prey and handled again by the incomplete mesenteries.
More recently identified is a primary means of consuming baceria that settles or is trapped against epitethial tissue, connective or otherwise: there are actually incomplete pores in the cell layer that collect and absorb bacteria
One the three kinds of cnidocyte "bags" (the cell structure that contains the thread or harpoon), spirocysts, have no harpoons, but sticky proteins on the thread that can help the polyp grip non mobile (non hostile) prey items (detritus, bacteria, etc.) that come in contact with the tentacles. Another major group of cnidocyte bags, of which most hobbyists are familiar, are nematocysts. This is a main way that larger, more motile prey items, such as crustacean zoea, megalops, mollusc trocophores, veligers, essentially most members of the zooplankton community and some members of the nekton community are captured. The prey is transferred using muscle process contraction in the tentacles, to the mouth, and are ingested using mucus sheets (some short-tentacled corals move the mouth to the captured prey). Larger prey items can be handled by extroverting the mesenterial filaments from the mouth and pre-digesting the prey via protease secretions and hydrolyzation.
For the most part, active feeding and digestive reactions/processes are pretty much "activated" by the presence of amino acids in the water from prey. In some species, circadian rhythm are created by the elevated presence of zooplankton at night. Relaxation/retraction during the day = reduced respiration and reduced need for energy.
Another feeding facet I am not very familiar with is bacterial cultivation by certain species. Horge or Jerel can likely give you a great explanation of how this works. I'd love to hear it myself

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So essentially, anything is fair game. Corals are playing catchup or balanceout with their energy requirements until a certain size/surface area is reached (what Jerel terms as "critical mass"). This size essentially where enough photosynthate and other resources are available due to high prey capture area and high zooxanthellae mitotic index/population density. This is usually when energy is budgeted to produce gonads along the mesenteries.
I don't have numbers on me (not at school, home for the holidays), but prey capture/consumption rate is very high per unit biomass on reefs, even with 70-90% of nutrition being derived from zooxanthellae. This prey capture rate must also be allowed to include the standard deviation that taking up nutrients and wastes of ectosymbiotic fish would provide (and farming bacteria, as well). For some species of arborose/ramose corals, this contribution to their nutrition can be quite significant. Therefore, corals are essentially bombarded with potential nutrition in the wild, and must use this fact to their every advantage.
BTW, I do remember that SPS, ramose corals such as
Acropora,
Pocillora and
Stylophora actually had the highest food rationing.
In conclusion: feed your corals. Detritus,
Artemia salina nauplii, copepods, fish poo, ground fish/shrimp/etc. proteins (blender mush!), etc.
I know I'm missing stuff... please jump in Horge!