The Rite of Separation was a Bajoran custom performed when two people ended their relationship.
The Rite involved celebrating for several days, giving the couple time to remember the good times and seek out "new opportunities". During this period they would remain cordial. The last part of the rite involved a simple ritual where the former couple would together recite words and break a bowl, in the presence of a witness.
Julian Bashir and Leeta performed the Rite of Separation on Risa when they ended their relationship in 2373. Worf thought it was a "frivolous" way to end a relationship, but Jadzia told him "not every relationship has to end like a Klingon opera". (DS9: "Let He Who Is Without Sin...")