Unicellular organisms eat using a processes called phagocytosis. This process allows the single-celled organism to bring food into the inner part of the cell for digestion.
A combination of chemicals on the surface of the single-celled organism as well as on the desired food allow the organism to recognize food and attach itself to the substance it is trying to eat. Once attached, the organism wraps its outer layer, which is called the outer cell membrane, completely around the food particle. Once the food has been engulfed by the outer cell membrane, the membrane pinches off towards the inside of the cell, creating a sac inside of the cell that contains the food called a vacuole. The vacuole and food together are called a phagosome.
The phagosome then comes into contact with a lysosome. A lysosome is a vacuole that contains enzymes. The lysosome then injects these enzymes into the phagosome, and once they are inside, the enzymes break down the food into individual molecules. When there are only individual organisms left of the food, it is then considered to have been digested or eaten. These molecules can then be used to fuel and build anything the single-celled organism needs.