During an average day, your body burns about 2,000 calories (when you are exercising heavily, it burns a lot more). That means that during waking hours, you are burning about 2 calories a minute. These 2 calories have the ability to raise the temperature of 1 kilogram of water 2 degrees C. If you weigh 50 kilograms (110 pounds), your body temperature rises one-twenty-fifth of a degree C (one-twelfth of a degree F) every minute. (For more information on calories and your body, see How Calories Work.)
Your body needs a way to dump that excess heat. If it doesn't, then body temperature rises into the danger zone in a matter of 30 minutes. Up to about 80 degrees F (24 degrees C), it's easy to dump excess heat simply through radiation (this is why air temperature "feels" comfortable at up to about 80 degrees F). Above 80 degrees F, your body does not have enough surface area to get rid of the heat fast enough, so your body turns on your sweat glands to make evaporative cooling possible.
Evaporative cooling works great if the air is dry. In high humidity, however, it doesn't work very well -- the sweat cannot evaporate because the air is already saturated with humidity. In high temperature/high humidity environments, your body can get into a dangerous situation where it cannot radiate or evaporate the heat away. The heat index that you see on the evening news is designed to make you aware of these dangerous situations.
The heat index takes the day's temperature and humidity into account and calculates what the temperature would be if the air were at 25-percent humidity or so (very dry). On this scale, high humidity can make you excruciatingly hot because your body has no way to eliminate excess heat. For example, 100 degrees F with 100-percent humidity is the equivalent of 195 degrees F at 25-percent humidity -- nearly the boiling point of water!
Here are some interesting links: