One's respiratory rate or breathing rate was a measurement of how many breaths they took within a given time frame.
A low respiratory rate was one of the indications of a tear of the middle meningeal artery. In 1986, Doctor Leonard McCoy questioned how a 20th century doctor would explain Pavel Chekov's symptoms, noting the patient's decreasing pulse, low respiratory rate and comatose state. The surgeon suggested fundoscopic examination, an option McCoy dismissed as unrevealing. (Star Trek IV: The Voyage Home)
In 2266, Christine Chapel reported to Dr. McCoy that surgical patient Joe Tormolen's breathing rate was dropping. He then ordered her to compensate for the change with the respirator. (TOS: "The Naked Time")