The majority of corals we see in captivity (with the exception of sun corals, a few soft corals- drendos, and a few gorgonians) contain symbiotic algae. This photosynthesises providing the coral with nutrients. In addition as far as I know, polyps can also capture food directly from the water column. The size of food which corals can capture obviously varies greatly- a frogspawn LPS coral is going to be able to consume a very different sort of food to a soft coral.
Generally in my experience as long as you have a healthy system with live rock, feeding most soft corals is unnessassary, although they may benefit from it occassionally with a good phytoplankton. Most LPS will benefit from a bit of meaty food occassionally, and I dont know about SPS.
Unfortunatly the most spectacular corals colour wise are often non-photosynthetic (since they are reflecting the light rather than absorbing it) and these will require heavy feeding.