Among many cultures, applause was a sign of respect and approval. The act of applause was also known as clapping.
Following a speech given by Minister Nathan Samuels announcing the formation of the Coalition of Planets, he received a round of applause. When Captain Jonathan Archer noticed his chief engineer, Trip Tucker's clap, was rather lackluster. After Archer pointed out to Tucker that "people were watching," he added "clap louder. That's an order." (ENT: "Demons")
In 2367, Reginald Barclay and Doctor Beverly Crusher performed a scene from Cyrano de Bergerac in the theater of the USS Enterprise. Despite an awkward and halting performance by Barclay, the crew, senior officers included, presented a round of applause and comments of approval for both Crusher and Barclay. Lieutenant Commander Data was confused at the applause for Barclay's "adequate" performance until First Officer William T. Riker explained that "it's polite." (TNG: "The Nth Degree")
Bajorans and Wadi applauded by clapping the back of one hand into the palm of the other. Cardassian applause was generated by tapping a table or armrest with a clenched fist. (DS9: "Move Along Home", "In the Cards", "Tribunal", "Sons and Daughters")
For Humans, applause was performed by clapping both palms together. This method was also used by non-Human Starfleet officers, and others within the Federation. In 2376, it was copied by the Qomar, who used it in appreciation of musical performances. This type of applause was also in use on Kelis' species in the Delta Quadrant. (VOY: "Virtuoso", "Muse")