Frostbite
One of the big problems with such cold conditions is losing body heat. Hypothermia is now fairly rare but it is still possible to get frostbite if you aren't careful.
Hypothermia
Hypothermia is when the body's core temperature (the temperature in the brain, heart and other organs) drops too low. It starts with intense shivering, then causes muscular weakness, difficulty in working, and a dull mental state that prevents the victim from realizing what is going on. Eventually the person loses consciousness, and the heart stops.
It is treated by preventing further heat loss, and then by slowly warming the person up again. It is rare to have anyone suffering from hypothermia these days in Antarctica, because people are trained to detect it and to prevent it happening.
Frostbite
A more common injury is frostbite, where body tissue actually freezes. It usually happens to fingers or to toes, where the blood supply is likely to slow down if the person gets too cold. The fingers (or toes) actually turn white, and become cold and numb, so the person cannot feel anything.
It is usually best to thaw out the tissue rapidly in warm water. But if the tissue has actually died, gangrene or poisoning can result. In such cases, the fingers or toes need to be amputated so that the gangrene does not spread to the rest of the body.
Efficient clothing is obviously very important in preventing both hypothermia and frostbite, and modern materials have made it far less likely that modern explorers will suffer from either.