(covers information from several alternate timelines)
A horga'hn was a statuette that served as a fertility symbol on Risa, and represented the Risian symbol of sexuality. It came in many sizes, from small carvings to statues to a hotel with a giant horga'hn built into it. It was believed that to own one was to call forth its powers. Displaying one publicly indicated that one was seeking jamaharon, a mysterious and pleasurable practice native to that world.
A stylized horga'hn was used as a door switch in several apartments on Risa in 2152. In addition, several cargo containers in the basement of the Risian nightclub were labeled with horga'hn symbols. (ENT: "Two Days and Two Nights")
In 2366, Commander Riker asked Captain Picard to acquire a horga'hn for him while the captain vacationed on Risa. An employee spotted the item and got the wrong idea about Picard. Picard didn't know until afterward that Riker was setting him up to appear that he was seeking jamaharon. (TNG: "Captain's Holiday")
On the way to Risa, Quark gave Dax, Bashir, and Leeta each a horga'hn, but made a point of not giving one to Worf. Once there, he eagerly tested the practice by displaying two horga'hn to a pair of Risian females, explaining when asked if he sought jamaharon, "I seek whatever you've got." They replied with the traditional Risian greeting "All that is ours is yours", to which Quark replied, "I'll take it!" Worf later threw his horga'hn against a wall in anger, shattering it. (DS9: "Let He Who Is Without Sin...")
In Naked Time, one of the holographic training drills given to the USS Cerritos crew in 2381, Steve Stevens was portrayed riding Jack Ransom and spanking him with a horga'hn while both were under the influence of polywater intoxication. (LD: "I, Excretus")