Right Side
As a general rule of thumb, the hurricane’s right front side (relative to the direction it is traveling) is the most dangerous part of the storm because the strongest winds and greatest risk from storm surge are usually found to the right of the center. Tornadoes are also more common here.
The winds are different between the right and left sides of the hurricane because of the combined effect of the hurricane wind speed and speed of the larger atmospheric flow (the steering winds). Looking at the graphic below, the right side is the eastern section of the hurricane. (If it were traveling east to west, the right side would be the north section.) The winds around the hurricane’s eye are moving counterclockwise. At Point A, the hurricane winds are nearly in line with the steering wind, which typically increases the strength of the winds. On the other hand, the winds at Point B are moving opposite those of the steering wind and therefore would be expected to be slower on this side. NHC forecasts take this effect into account in determining official wind estimates. The wind value provided is the strongest wind anywhere in the hurricane's circulation.
Note, however, that this is just a general rule. Some hurricanes have features that can have equal or stronger damaging winds on the left side of the storm. Also, because of uncertainties in forecast track, communities on the left side of the forecast track need to plan for the full brunt of the storm.