Lynn Margulis proposed this theory in 1996. It explains how the eukaryotic cells engulfed through endophagocytosis some organelles such as mitochondria and chloroplasts, which evolved from bacteria. The cell and the mitochondria/chloroplast entered then in a endosymbiotic relationship: by one side, the new organelle (the bacterium) provided the cell with energy, while the cell acted as a host for the bacterium.
According to this theory, probably the cell was initially anaerobic, while the bacterium was aerobic. The cell failed in digesting the bacterium, which lived inside the cell, producing energy that the cell could use and allowing it to breath aerobically. Eventually the bacterium could not live independently, becoming a cell organelle.