(covers information from several alternate timelines)
A wedding was a ceremony that in some form or another had been independently created on many humanoid homeworlds. It was the act of marrying two or more people depending on culture often for the purpose of religion and/or sexual reproduction.
In the Federation, particulary in Starfleet might be performed by Federation or Starfleet officials. James T. Kirk officated the wedding of Robert Tomlinson and Angela Martine, though the ceremony was interrupted by a confrontation with a Romulan ship in which Tomlinson was killed. (TOS: "Balance of Terror"). Jean-Luc Picard married Miles O'Brien and Keiko Ishikawa in 2367. (TNG: "Data's Day") Sisko performed the ceremony for Ensign Hoya, as well as Rom and Leeta's wedding in late 2373, though . (DS9: "The Ship", "Call to Arms") William Ross performed Benjamin Sisko and Kasidy Yates' wedding in 2375, although a minister was preferred by Yates' mother. (DS9: "'Til Death Do Us Part", "Penumbra") Kathryn Janeway married Tom Paris and B'Elanna Torres, as did their demon planet biomimetic copies. (VOY: "Course: Oblivion", "Drive")
Deanna Troi and William T. Riker were married in Alaska in 2379 according to Human wedding traditions. They would later go to Betazed to experience the traditions of the Betazoid wedding. (Star Trek Nemesis)
In an alternate timeline, Harry Kim was the best man at the wedding of Tom Paris and Kes. He was supposed to give the toast, but was so nervous that he ended up spilling champagne all over Paris's dress uniform. (VOY: "Before and After")
Both Sisko's and Admiral Ross' wedding dialog were written to pay homage to Kirk's dialog in "Balance of Terror".