Dry, warm weather in October or November. Traditionally the phrase is applied to warm weather after the first frost of autumn.

In San Francisco, which doesn't get frost until winter (if at all), the phrase can refer to the annual heat wave at the end of September/beginning of October, where grateful residents enjoy the 90 degree plus temperatures which follow the foggy, cold summer of the city's Mediterranean climate.

First usage is cited in America in a 1778 letter by Frenchman St. John de Crevecoeur:

"Sometimes the rain is followed by an interval of calm and warmth which is called the Indian Summer; its characteristics are a tranquil atmosphere and general smokiness. Up to this epoch the approaches of winter are doubtful; it arrives about the middle of November, although snows and brief freezes often occur long before that date."
Why "Indian" summer? Lots of contrasting theories, many involving Native American seasonal hunting, as well as the loading of ships to cross the Indian ocean during this fair weather time of year.

Sources: http://www.crh.noaa.gov/dtx/i-summer.htm;
also going outside, licking my finger, and holding it up and squinting skyward