In more detail
The US National Weather Service computes it with the Rothfusz regression, a polynomial in temperature (°F) and relative humidity fitted to Steadman's model of human heat balance, with small corrections at very low and very high humidity and a simpler formula below about 80 °F. The idea is that humid air slows the evaporation of sweat, the body's main way to cool down. Values assume shade and a light wind; full sunshine can add up to about 15 °F.
Why it matters
The NWS uses the heat index for its warning categories: 80 to 90 °F caution, 90 to 103 °F extreme caution, 103 to 124 °F danger and 125 °F or more extreme danger, as the risk of heat exhaustion and heat stroke rises. It guides when to reschedule outdoor work or sport, drink more and check on older people and children. Other measures exist, such as the Canadian humidex and the wet-bulb globe temperature used for work and sport, and their numbers are not interchangeable.
Example
90 °F (32.2 °C), 70 % RH → feels like 105.9 °F (41.1 °C) danger
90 °F (32.2 °C), 40 % RH → feels like 90.7 °F (32.6 °C)
95 °F (35.0 °C), 50 % RH → feels like 105.2 °F (40.7 °C)
Common confusion
The heat index is the hot-weather counterpart of Wind chill; neither changes the actual air temperature. Humidity can also be expressed as the Dew point.