All life on Earth must sustain itself by producing or consuming energy. Many organisms such as plants and algae produce energy, but the subsequent parts of the food chain involve consumers that undergo some process of cellular respiration in order to break down energy that came from the previous producers.
Cellular respiration is a type of metabolism in living organisms that releases energy in the cell. These reactions are said to be catabolic, which means they are a set of pathways that require larger molecules to be broken down into smaller molecular units so ATP can be produced. ATP provides energy throughout the organism.
Another way to think about it is these kinds of organisms are heterotrophs, meaning they cannot make their own food. Autotrophs like plants, algae and many single-celled organisms can make their own food, utilizing energy from the sun to produce sugars in the photosynthetic process. In other words, heterotrophs break molecules down instead of build them up. Respiration is just a way of accomplishing this.
Aerobic respiration, which is respiration that requires oxygen, is utilized by some eukaryotes. A eukaryote is an organism that contains organelles within its cells. These organelles are able to take advantage of more complex metabolic pathways that require oxygen in order to produce greater amounts of energy. Some eukaryotes like plants use photosynthesis instead, but all animals are eukaryotes that utilize aerobic respiration.
Another type of respiration is anaerobic respiration. This is a more simplified process compared to aerobic respiration and uses an inorganic compound in place of oxygen. However, it cannot produce quite as much energy, so it is mostly found in prokaryotes, or organisms lacking organelles, like bacteria.
Fermentation is a similar process that breaks sugar down into carbon dioxide and ethanol or sometimes even lactic acid (which is an acid often metabolized in muscle cells instead of oxygen, manifesting itself as a burning sensation once work begins to outpace energy production). Fungus such as yeast utilize fermentation for energy production.