The main way amphibians deal with winter's cold temperatures is to hibernate, or to "go dormant." Different amphibian species hibernate in different ways.
When cold weather comes many kinds of frogs swim to the bottoms of ponds and lakes where they rest on the bottom or partially burrow into the mud. Frogs have lungs and breathe air through nostrils, so why don't they drown when they stay on a lake's bottom all winter? It's because their highly specialized skin allows them to absorb oxygen through their skin, and giving off carbon dioxide the same way.
Amphibians overwintering in the bottom of ponds must be deep enough that they will not be frozen into solid ice, plus the water must have an adequate amount of oxygen. Sometimes the oxygen level at the bottom of ponds drops so low that "winter kills" of frogs result.
Amphibian bodies when they hibernate do continue to burn energy, but their metabolic rates -- the rates at which they burn energy -- slow down so much that they don't burn much. Many species can survive underwater for months, their bodies very slowly burning fat stored in their bodies. Usually frog fat bodies are located just inside the abdominal wall.
Sometimes during the winter after a few unusually warm days hibernating amphibians will become active and you may even hear frogs croaking. Sometimes the aquatic salamander called a mudpuppy moves about beneath the ice and ice-fishermen sometimes catch them!
Sometimes amphibians such as toads hibernate on land, digging below the frost line to avoid freezing. Toads where winters are especially hard may dig down three feet or more! Certain salamanders may also hibernate deep below the ground's surface, but they can't dig the way toads can, so they must search for abandoned burrows or other natural holes.
Some frogs in the far north can actually freeze solid. When cold weather approaches they just burrow under the forest floor's leaves and debris, and when freezing weather comes much of the water in their bodies goes out (more than half), its veins fill with an antifreeze-like mixture of sugars and sugar alcohols, and then the somewhat-dried-out frog simply freezes. While it's frozen, it's hard, and ice forms around the frog's organs. However, the frog's individual cells remain unfrozen and intact. Of course when it's frozen it doesn't breathe nor does its heart beat. Brain activity is immeasurable.
Amphibian pets kept warm indoors usually don't need to hibernate. However, some species need a period of dormancy before they can reproduce.
Hot, dry weather also can be stressful for amphibians, and often during such times they may go dormant. However, now instead of being called "hibernation" it's called "estivation," sometimes spelled "aestivation. The word "estivation" is derived from the Latin aestas, which means summer, same as the term "hibernation" is derived from the Latin hiberna, which means "winter."
When some amphibians estivate they move underground where it is cooler and more humid. During estivation an amphibian's breathing, heart rate, and metabolic processes such as digestion all dramatically slow down. This decreases the organism's need for water. Some frogs and salamanders form a mucusy cocoon around themselves to prevent water loss through their skin. When rains return, estivating organisms become active again -- they "break their dormancy" or "end their estivation."