Insects breathe through spiracles on the outside of their exoskeletons, which link to a network of oxygen-delivering tracheal tubes. These tubes end in liquid-filled tracheoles where oxygen dissolves before diffusing to the cytoplasm of adjoining cells.
Without the aid of lungs to move air in and carbon dioxide out of their bodies, insects must use their spiracles to perform this respiratory function. Most insects have one pair of spiracles on each body segment: head, abdomen and thorax. Smaller insects are oxygenated entirely by passive diffusion, while larger ones must alternately open and close spiracle valves to draw air in and force carbon dioxide out through contractions, essentially turning their whole bodies into a semblance of a lung.
The tracheal tubes serve the same function that hemoglobin-carrying arteries serve in larger animals; the critical difference is that it is air, not blood, that is carried. The presence of this high volume of air allows insects to float more easily, and many insects use it as a reserve of oxygen while underwater.
It is only because of insects' tiny size that this respiratory apparatus works; a human-sized insect would not be able to get fresh air to the center of its body.