Eating, or ingestion, is one of the primary qualifiers that describe a living organism. The most common form of eating of mono-form corporeal lifeforms involves ingesting organic material, which is chemically broken down by the organism's digestive system and distributed to its cells for minerals and energy.
On the Enterprise-D group of parasitic organisms attempted to consume the crews' cellular peptides (which would have caused them to collapse into pile of chemicals and thus die). However, they were discovered and purged from their bodies. (TNG: "Phantasms")
Some lifeforms, such as the Crystalline Entity, also consumed material, but converted it to energy directly rather than chemically breaking it down. (TNG: "Silicon Avatar") Some organisms fed on energy and either required a host, such as Ronin (TNG: "Sub Rosa") or fed on it directly, such as Farpoint Station.
Cultures of some species, such as the Kreetassans considered eating in public akin to mating in public. (ENT: "Vox Sola")
While observing the giant space amoeba discovered in 2268, Doctor Leonard McCoy described it as a very simple form of life capable of displaying almost all the functions living organisms perform, including the ability to eat, breathe and reproduce. McCoy, however, was not sure what it ate, but Spock speculated that perhaps it consumed energy itself. (TOS: "The Immunity Syndrome")