A healer was a title for one who practiced medicine, equivalent to an Earth doctor. The term was used by Klingons as well as by Vulcans, the latter of which also applied it to veterinarians. (ENT: "Affliction"; TAS: "Yesteryear")
Klingon doctor Antaak was disowned by his father when he decided to become a healer. Martok, on the other hand, offered Dr. Bashir to include his deeds as Worf's healer when asking Keedera himself to write a song about his glorious fight against the Jem'Hadar. (ENT: "Affliction"; DS9: "By Inferno's Light")
Senva, from the Vulcan Medical Institute, was a prominent healer, nominated for the Carrington Award in 2371. (DS9: "Prophet Motive")
This title apparently does not apply to all Vulcans who practice medicine, as Fer'at in ENT: "The Expanse" and the Vulcan doctors in ENT: "Stigma" used the title of Doctor.
Some Humans have also used the title of healer to describe a person who has medical training, often self-taught, but may not have the formal education of a doctor. Felisa Howard was a healer on Caldos colony until her death in 2370. Her granddaughter Beverly Crusher, while under the influence of the anaphasic lifeform called Ronin, temporarily resigned her Starfleet commission to follow in her grandmother's footsteps. (TNG: "Sub Rosa")
The USS Pasteur's dedication plaque quoted part of a version of the Hippocratic Oath which referenced Apollo as a healer. (TNG: "All Good Things...")
Dr. Julian Bashir as well as The Doctor also used the term healer sometimes to refer to a medical doctor. (DS9: "The Passenger"; VOY: "Critical Care")
In a deleted scene from the Star Trek: The Next Generation second season episode "Up The Long Ladder", Brenna Odell calls herself a healer and midwife. [1]