Breathing was a physiological process of gas exchange that remains one of the primary qualifiers that describe a living organism.
While observing the giant space amoeba discovered in 2268, Doctor Leonard McCoy described it as a very simple form of life capable of displaying almost all the functions living organisms perform, including the ability to breathe, eat and reproduce. (TOS: "The Immunity Syndrome")
For most humanoids breathing involves the intake of oxygen (respiration) and the exhale of carbon dioxide, to and from the lungs. The inability to breathe is known as suffocation. The predecessors of the Aquans, known to their descendants as "air-breathers", developed technology, a surgo-op process, that allowed their species to mutate into "water-breathers". (TAS: "The Ambergris Element")
A low respiratory rate was one of an injured Pavel Chekov's symptoms in 1986. (Star Trek IV: The Voyage Home)
Some individuals stopped breathing while they slept. In 2372, Quark explained that he used an eavesdropping device to monitor a sleeping Rom, who sometimes stopped breathing during sleep. Odo didn't believe him. (DS9: "Crossfire")
Not all humanoids breathed the same substances, for example:
- Axanar breathed nitrogen-methane. (ENT: "Fight or Flight")
- Barzans wore breathers. (DIS: "Rubicon", et al.)
- Benzites wore a breathing apparatus in order to breathe in an oxygen-nitrogen environment. (TNG: "Coming of Age", "A Matter Of Honor")
- Lorillians, prior to the age of four, were only able to breathe methyl oxide. (ENT: "Broken Bow")
- Lothra breathed hydrogen. (DS9: "Melora")
- Tesnians required at least six hours of boron gas per day. (ENT: "Shuttlepod One")
- Yalosians breathed a corrosive mixture of 60% nitrogen, 10% benzene, and 30% hydrogen fluoride. (DS9: "Improbable Cause")
- Zaranites breathed fluorine and could not survive in oxygen, requiring them to wear breathing masks in class M environments. (Star Trek: The Motion Picture)
A one-word instruction to "breathe", which T'Pol says to Tucker at the end of a neuro-pressure scene in ENT: "Similitude", wasn't scripted.