Prokaryotes may perform aerobic (oxygen-requiring) or anaerobic (non-oxygen-based) metabolism, and some can switch between these modes.
Obligate aerobes is the mechanism in which Prokaryotes that need oxygen in order to metabolize. These prokaryotes perform aerobic cellular respiration, a process that uses an electron transport chain to extract energy from fuel molecules. Oxygen is needed because it is the final electron acceptor at the end of the electron transport chain.
Obligate anaerobes is the process in in which Prokaryotes that can't tolerate oxygen then only perform anaerobic metabolism. C. botulinum, the bacterium that causes botulism (a form of food poisoning) when it grows in canned food, is an obligate anaerobe so it multiplies well inside of sealed cans by using fermentation, which is a process that converts pyruvate to either lactic acid or ethanol. Alternatively, they may carry out anaerobic respiration, cellular respiration that uses a non-oxygen molecule (such as nitrate or sulfate) as a final electron acceptor in the electron transport chain.
Facultative anaerobes use aerobic metabolism when oxygen is present, but switch to anaerobic metabolism if it is absent.
Cellular respiration takes place in the mitochondria which are present in eukaryotic cell but not present in prokaryotic cell as these cells are diploid of cell organelles and in this region of the cell the energy (ATP) is produced in the cell. Eukaryotes cannot tolerate without oxygen so it do not perform anaerobic metabolism because anaerobic metabolism takes place without oxygen.
So the above are the reasons that why prokaryotic exhibit anaerobic and aerobic respiration and eukaryotic exhibit aerobic respiration only.