A breathing mask was a special device 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")
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")
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 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")
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)