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")
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.