I don't have any fans, apart from an electric one
The meaning varies. AFAIK in the Thai tradition it's an official sign of the monk's status as being eligible to grant ordination to others.
In the Vinaya, a fan was allowed to shield the heat of the sun, since a parasol is not permitted. The monks carry a fan on almsround.
In Burma, the fan may also confer some official status but it may also be used simply for its practical purpose of cooling the body in hot weather. While preaching, the fan may be held in front of the monk's face to avoid distraction.