A cell is the basic structure of the body. Organelle means tiny organs and these structures within a cell perform specialized functions. For example, the nucleus contains all the DNA and directs the synthesis of proteins and the process of cell replication. Certain organelles play a role in breaking down compounds such as proteins or destroying bacteria, for example. Thus, they have a digestive function.
According to "Molecular Biology of the Cell" by Bruce Alberts, the cell is surrounded by a membrane, or outer layer, that protects it and thus is analogous to skin. The cell is filled with fluid in which the organelles and cellular structures are located. The fluid is called cytoplasm. It contains some digestive enzymes that are responsible for regulating the environment of the cell. However, the bulk of digestive enzymes are sequestered into specialized organelles so that they do not damage the cell.
Lysosomes, which are organelles that are surrounded by their own membrane, have an acidic interior. They digest waste in the cell and play a role in phagocytosis. Phagocytosis is basically the way a cell "eats" things in its environment. The cell membrane surrounds the substance, such as a bacterium, and traps it in a small spherical section of membrane in the cytoplasm called the phagosome. Lysosomes then fuse to it to digest bacteria, viruses and other substances. Within the cell, the lysosomes can digest non-functioning organelles, food particles and so forth.
Peroxisomes are also surrounded by a membrane. They produce hydrogen peroxide and similar chemicals that use oxygen to break down chemicals. They play a role in many reactions, especially breaking down fatty acids for energy in the cell.
The mitochondria are mainly are involved in energy production in the cell. They use oxygen to synthesize ATP, or adenosine triphosphate, which stores energy. They are not digestive organelles, but they play a role in the process of apoptosis. Apoptosis is programmed cell death, or cell suicide. If the cell becomes overly mutated or develops too many problems, it can program its own death to save the organism, for example, in an attempt to prevent cancer. According to "Robbins and Cotran Pathologic Basis of Disease" by Dr. Vinay Kumar, the mitochondria plays a role in this by activating enzymes called caspases that help digest cellular materials.