A fan club was a group of people showing great respect or admiration for a certain person, group or cause.
When Nick Locarno saw Beckett Mariner heading towards Nova Squadron in 2368, he warned Sito Jaxa that her (one cadet) fan club was approaching. (LD: "Old Friends, New Planets")
While playing the role of Chef in a holoprogram in 2370, William T. Riker told Charles Tucker III he sounded like a lifetime member of the Jonathan Archer fan club. (ENT: "These Are the Voyages...")
After attending a conference on the occupation of Bajor in 2373, Garak stated that Odo seemed to have quite a fan club on Bajor due to the attention Odo attracted. (DS9: "Things Past")
A group of Qomar visited the USS Voyager in 2376 and acted like a fan club of The Doctor. They also attended a concert he gave. (VOY: "Virtuoso")