Breathing and respiration are often used as synonymous. But it is not correct. Breathing as a separate process is noticed only in few organisms (animals), while respiration occurs in all organisms.
Further, breathing is a physical process, while respiration is a chemical process taking place in all the cells. Breathing may be regarded as the last step found only in aerobic respiration, where atmosphere oxygen participates. There can be respiration without breathing, but the reverse is not possible.