Final answer:
The mitochondrion is the vital organelle for aerobic respiration in eukaryotic cells, consisting of two membranes and hosting processes like the Krebs cycle and electron transport chain to produce ATP.
Step-by-step explanation:
The organelle necessary for aerobic respiration in eukaryotic cells is the mitochondrion (mitochondria, plural). These are the sites of cellular respiration, generating ATP, which is the energy currency of the cell. Mitochondria are unique in having their own DNA and ribosomes similar to prokaryotic organisms, which supports the endosymbiotic theory that they were once free-living bacteria. The key processes such as the Krebs cycle and electron transport take place in and across the mitochondrial membranes.
Each mitochondrion is encased by two lipid membranes, with the inner membrane folded into cristae providing a large surface area for the electron transport chain and ATP synthesis. The number of mitochondria within a cell correlates with the cell's energy needs, evident in energy-demanding cells like muscle cells that contain many mitochondria. Organisms performing aerobic respiration require oxygen as the final electron acceptor, contrasting with anaerobic respiration, which utilizes other final electron acceptors.