A frog breathes through its skin, the inner surface of its mouth and its lungs, depending on its circumstances. When their skin is moist, and particularly when they are in water where it is their only form of gas exchange, they breathe through their skin. When it is necessary, such as on land, they take air into their lungs by pushing it from their mouths with their nostrils closed.
Frogs lack a diaphragm or rib muscles to expand their thoracic cavity and thus take in air. Thus, they breathe into their lungs opposite to how mammals do, using positive pressure to inhale and negative pressure to exhale. They do this by lowering the floor of their mouths to draw in air from the outside, and use the same process to draw the air out of their lungs.
Frog lungs are fairly inefficient, however, and a frog's metabolism is relatively slow, being cold-blooded. When at rest, frogs use their lungs only rarely, instead relying on their skin and their inner mouth surface, which is quite permeable to oxygen, for gas exchange. Their skins are thin and membranous, and are permeable to both water and gases. Frogs do not even develop their lungs until adulthood.