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")