Tracheotomy

A tracheotomy is an emergency medical procedure used on humanoids who have an obstruction in their throat that prevents breathing.

Specifics
In this procedure, an incision is made about an inch below the cricoid cartilage in the patient's trachea. 

On Earth, the procedure had been used for millennia. The modern surgical procedure was codified by Chevalier Jackson in the 20th century. 

History
In 1931, in a timeline created when Leonard McCoy saved the life of Edith Keeler, McCoy saved the life of Danny Johnson in Hayden, South Carolina by performing a tracheotomy. Unfortunately this revealed his medical expertise. When Doctor William Lyles inquired where McCoy earned his credentials, McCoy realized that he couldn't cite the University of Mississippi because its School of Medicine hadn't been founded yet. 

A tracheotomy aboard the Enterprise saved the life of Ensign Hawkins Young. Engineer Galatea Kostas performed the procedure, thanks to having taken some Starfleet Academy basic medical classes along with training in sickbay. 

McCoy's first combat casualty occurred during a scouting mission on a planet, when a Klingon attack severely injured a nearby lieutenant. McCoy tried to give him a tracheotomy to help him breathe, but he died before the procedure was completed. (SS TOS (Safe Harbors))

In 2269, McCoy performed a tracheotomy on Huff, a security guard who had been shot in the throat with an arrow by a tribe of young Arivne on the planet Anomaly. When an electronic scalpel failed to activate, McCoy had no choice but to use a vibroknife covered with dirt and tree sap. He asked Nurse Christine Chapel to clamp the carotid artery, and then Huff was put on a stretcher for immediate surgery. 

In 2367 in the mirror universe, after $𝜇$ tried to stab Jean-Luc Picard, an injury to Picard's neck was treated in sickbay by $𝜇$. She commended that the wound would have looked like an amateur tracheotomy had it cut deeper. She repaired the damage with a protoplaser. 

In 2375, McCoy related the experience of his first combat casualty, including the tracheotomy, to Commander Julia Rivera during a combat situation. (SS TOS (Safe Harbors))