To respire is to breathe in and out. After a calf is born, a farmer might watch it respire for a while to make sure it's okay.
While you can use the verb respire simply to mean "breathe," it's most often used in a medical or scientific context. A nurse might worry about the rate at which a patient respires, and a biologist might discuss the way a plant respires at night, when light doesn't reach its leaves. The Latin root, respirare, means "breathe again" or "breathe in and out," from re-, "again," and spirare, "to breathe."