Jellyfish don't breathe in a typical fashion, as they have no respiratory system; their ultra-thin membranes are able to diffuse oxygen into their cells from the water around them. Because of their physical makeup, jellyfish actually require much less oxygen than many other sea creatures.
The structure of a jellyfish includes three layers; an outer epidermis, an inner gastrodermis and a middle layer called the mesoglea, which is very elastic and similar to jelly--hence the name. The middle layer does not require any oxygen to survive, which is why the jellyfish's oxygen requirements are so low. In addition to the ease in which they acquire oxygen and their low requirements for it, the mesoglea layer also can store oxygen, which allows jellyfish to thrive in conditions that are oxygen poor.