Combustion- Combustion is a chemical reaction that burns fuel and an oxidant (a compound that transfers oxygen atoms or gains an electron) and produces heat and/or work. It is an exothermic reaction because it releases energy, which usually occurs when the oxidant burns away as a natural gas like methane and then binds with what is left. Fire is commonly thought of when it comes to combustion, but it is more accurate to say the initial products burn away. Respiration- Cellular respiration also is called oxidative metabolism because it uses a set of chemicla reactions to change the oxidiative state of the initial products. In animals and simpler organisms, respiration involves glucose and oxygen. The oxygen does not come into play until the end, when free electrons need something to which to bind. Carbon-Di oxide and water are the bi-products of the reaction.
Tarabarbie gave the long answer. The short answer is they are basically the same thing: reaction of organic compounds with oxygen to produce free energy, CO2 and water. Respiration uses enzymes to complete the chemical reaction at a low temperature and a lower rate.