Breathers (also called life masks) are a type of life-support gear worn by Viper pilots (TOS: "Lost Planet of the Gods, Part I") and people who need to operate in different atmospheres (TOS: "The Gun on Ice Planet Zero, Part I").
Boomer and the others trapped in the rejuvenation center use breathers to breathe despite the presence of the intense smoke. They later use the breathers to survive the vacuum when Adama's plan to extinguish the fire is put into action (TOS: "Fire in Space").
Malfunctioning breather gear causes the operator to suffer from asphyxia. As Jolly is landing on Galactica, Boomer assumes that his poor approach and subsequent landing are caused by his breather gear "leaking carbide" (TOS: "Lost Planet of the Gods, Part I").