According to a national study, most people believe that the cone represents where the winds from a tropical storm or hurricane will impact. It actually represents the probable track of the center of a tropical cyclone based on the errors of the past five years. The National Hurricane Center plots where it thinks the storm will be in 12, 24, 36 hours etc. and places those dots on a map.
This image is of Tropical Storm Don on Friday morning. The dots are the forecast from the NHC, but the cone represents the how far the center of the storm can be away from the forecast point.
Notice the circles in this image. The size of each circle is set so that two-thirds of the five-year historical forecast centers fall within the circle. The red dots out of the track are the past position of Hurricane Danielle in 2010. The cone is created by connecting the circles. In the cone, you can see how the 12- and 24-hour forecasts were just a touch off. The center of Danielle was faster and to the right of the circles, but still in the cone. But the forecast was off big time four and five days out. The center of the storm wasn’t anywhere near the cone because the storm went a lot faster than expected. What I always tell people is the forecast cone is good three days out, but four or five days a lot can change. Danielle is a good example of the 1/3 of the time being outside the forecast cone.