(covers information from several alternate timelines)
An Aamaarazan wearing a breathing mask
An oxygen mask being worn by the head guard of a trellium mining colony
A breathing mask was a type of breathing apparatus worn over the face to allow species that did not breath the local atmosphere to breath their own supply. They were sometimes also used for medical purposes to supply a patient with special breathing gas.
The alien foreman of a trellium mining colony had to breathe with the help of a mask because the toxic fumes had destroyed his lungs. His workers (as opposed to the miners) of his species also wore similar masks, presumably as a preventive measure. (ENT: "The Xindi")
During a simulation in 2153, an oxygen mask was used by Jonathan Archer and Degra in order to recover from reactor coolant which leaked into a simulated Malosian cargo shuttle they were aboard. (ENT: "Stratagem")
This device was called an "oxygen mask" in the final draft script of "Stratagem".
In 2154, Persis distributed life-support masks to a group of Augments she was in and to their "father", Arik Soong. Persis was mindful that there weren't enough masks for them all, but Soong told her they didn't need masks. (ENT: "Cold Station 12")
These devices were called "life-support masks" in the final draft script of "Cold Station 12" and were referred to simply as "masks" on screen.
At early talks for a Coalition of Planets, a Coridanite ambassador wore a breathing apparatus over most of his face. (ENT: "Demons")
In 2155, breathing masks were worn by an away team who traveled, in a shuttlepod, from Enterprise NX-01 to the surface of Mars. Though Phlox briefly tried his mask on shortly prior to the mission, every member of the away team donned one of the masks once the shuttlepod had landed on the planet. They continued to wear them while journeying across the planet's surface, then took them off inside an airlock in Mars' verteron array. (ENT: "Terra Prime")
In the final draft script of "Terra Prime", these masks were initially described as "small re-breather units resembling fighter pilot's breathing masks". They were thereafter repeatedly referred to as both "re-breather masks" and simply "masks".
In 2259, Captain Christopher Pike had to wear an oxygen mask while visiting a planet in the Vaultera Nebula. (SNW: "Ad Astra per Aspera")
In 2263 of the alternate reality, Montgomery Scott wore a breathing mask while inside an advanced long-range torpedo he fired from the USS Enterprise into space and then piloted to the surface of nearby planet Altamid. (Star Trek Beyond)
In 2266, an oxygen mask was worn by Simon Van Gelder when he arrived as a stowaway aboard the USS Enterprise, beamed there in a box containing some research documents. As he climbed out of the box, he took off the mask and left it in the container. (TOS: "Dagger of the Mind")
This device was identified as an "oxygen mask" in the final draft script of "Dagger of the Mind".
Doctor McCoy and Lieutenant Jaeger wearing breathing masks
Upon beaming down to the planet Gothos, a face mask was worn by every member of a landing party consisting of Doctor Leonard McCoy and Lieutenants Jaeger and DeSalle, as they expected the planet's surface to be stormy and virtually unbearable for life without oxygen and life support systems. The team took off their masks as they found the surface of Gothos was far different and comfortable than they had detected, with an atmosphere which was exactly the same as their own. (TOS: "The Squire of Gothos")
In the scripts of "The Squire of Gothos", these devices were referred to as "face masks".
Zaranites typically wore special breathing masks. Such a mask was worn by each of at least three Zaranites who were serving aboard the USS Enterprise during the V'ger incident in the 2270s. (Star Trek: The Motion Picture)
Behind the scenes, the Zaranite breathing masks were intended to be because the Zaranites breathed fluorine and couldn't survive in oxygen, making it necessary for them to wear the masks on Earth. (The Making of Star Trek: The Motion Picture, p. 130)
Other breathing masks were worn by Aamaarazan Federation councilors at the trial of James T. Kirk in 2286. (Star Trek IV: The Voyage Home)
According to the non-canon reference book Star Trek IV Sourcebook Update, the props for the Aamaarazan breathing masks were actually recycled Zaranite masks from The Motion Picture.
Janeway and Kim wearing breathing masks
While performing repairs inside a class 9 nebula, the crew of the USS Voyager was forced to wear breathing masks with converters due to a malfunction of the ventilation system. (VOY: "Year of Hell, Part II")
The call sheet for the day of filming listed these devices worn by Janeway and Kim as "facemasks with converters".
In the writers' second draft script of ENT: "Breaking the Ice", Malcolm Reed wore "an oxygen mask" after a tear in his EV suit caused him inhalation problems, though no such mask appears in the final edit of the episode. Similarly, as evidenced by the final draft script of ENT: "Similitude", Charles Tucker III was to be wearing "an oxygen mask" over his mouth and nose. In the final version of that episode, though, he is shown with only a thin tube that runs next to his nose. Similarly, Hoshi Sato was scripted to be wearing an "oxygen mask" shortly prior to her death in ENT: "Observer Effect", though no such device appears in the final version of that episode.