Single cells such as Amoeba absorb food particles from their environment and digest them directly inside themselves (their single cell).
There is no designated area of the cell where phagocytosis takes place as the food gets absorbed. Pseudopods can extend out of the cell to aid with capturing the food.
During absorption, vesicles within the plasma membrane of Amoeba can be formed to imbibe the food particles.
Digestive enzymes inside the phagolysosome break down the food into smaller particles, and excretory byproducts are released back into the environment.
Once an organism is multicellular, it can concoct actual structures dedicated to digestion. The most basic such structures involve simple, undifferentiated sac-like guts with just a single opening, while more complex ones are tubes with two openings (one for ingestion and one for egestion; so the nutrients in food and the waste products don’t mix together) and separate regions for digesting different types of food substance.
Hydra is an example of an organism with multicellular, simple digestion.
Its tentacles help it catch prey, such as Daphnia, which is then ingested via the mouth into its gastrovascular cavity. Cells in the gastrodermis secrete digestive enzymes such as those that break down protein, and the prey gets digested inside Hydra. Waste products are eliminated (egested) through the same mouth.
The more complex multicellular digestive systems, including the human digestive system, have separate openings such as the mouth and the anus, as well as specialised regions. For example, the small intestine contains three different sections: one breaks down proteins and lipids, as well as neutralises the acidity coming from the stomach; one carries out the absorption of broken down nutrients into the bloodstream; while the third section absorbs bile acids and vitamin B12, alongside any remaining nutrients.
More on the human digestive system coming up.