Insects require oxygen to live, and produce carbon dioxide as a waste product, just as we do. To say insects breathe, though, might be a stretch. They don't have lungs, nor do they transport oxygen through their circulatory systems. Instead, insects use a series of tubes called a tracheal system to perform gas exchange throughout the body.
Gas exchange, or what we think of as breathing, is accomplished mostly by simple diffusion through the cell walls. Air enters the spiracles, and moves through the tracheal system. Each tracheal tube ends in a moist tracheole, a specialized cell for exchanging gases with another cell in the body.
When air reaches the tracheole, oxygen dissolves into the tracheole liquid. Through simple diffusion, oxygen then moves to the living cell and carbon dioxide enters the tracheal tube. Carbon dioxide, a metabolic waste, exits the body through the spiracles.
This explains the movement of gases, but can insects control their respiration? Yes, to some degree. The insect opens and closes the spiracles using muscle contractions. An insect living in a dry, desert environment will keep the spiracle valves closed to prevent moisture loss. Insects can also pump muscles throughout their bodies to force air down the tracheal tubes, thus speeding up the delivery of oxygen. In heat or under stress, insects can even vent air by alternately opening different spiracles and using muscles to expand or contract their bodies.
Still, diffusion places some limits on the insect body. The rate of gas diffusion cannot be controlled, and only proves efficient for small organisms. This limiting factor probably benefits us, as otherwise we might find ourselves living with giant-sized insects! As long as insects breathe using simple diffusion, they aren't likely to get much larger than they are now.