A hug or embrace was the wrapping of one's arms around another as a sign of affection or act of comfort. It was often done upon meetings and partings.
When Wesley Crusher's dad left, he hugged young Wesley goodbye. Wesley still recalled how his father looked as of 2366.
In 2366, a Koinonian recreation of Marla Aster offered Marla's son Jeremy Aster a hug, which he accepted after a moment. (TNG: "The Bonding")
In 2367, while consoling Deanna Troi over the (ultimately temporary) loss of her empathic abilities, William T. Riker embraced her. When she asked if he handled all his personnel problems thusly, he replied, "Sure. You'd be surprised how far a hug goes with Geordi or Worf." (TNG: "The Loss")
Later that year, surprise visitor Q offered Jean-Luc Picard a big hug. (TNG: "Qpid")
In 2368, Lwaxana Troi sought a hug from Alexander Rozhenko. (TNG: "Cost of Living")
In 2370, Jadzia Dax offered Kor a hug upon their reunion (having previously known him as Curzon Dax), rather than the kiss he requested. (DS9: "Blood Oath")
In 2371, Benjamin Sisko asked his son Jake Sisko for a hug, which Jake declined. (DS9: "The Abandoned")
Also that year, Vice Admiral Patterson "ordered" "Katie" Janeway to give him a hug. (VOY: "Relativity")
In 2373, Miles O'Brien suggested that his daughter Molly tell her mother Keiko what happened to her bonsai tree after she hugged and kissed Keiko. Molly declined to deliver the bad news. (DS9: "The Assignment")
Later, Worf preemptively asked Julian Bashir not to hug him. (DS9: "Let He Who Is Without Sin...")
In 2374, Quark, recovering from his brief stint as the female Ferengi "Lumba", asked Odo to give him a small hug as comfort for his raging emotions and hormones. Odo acquiesced. (DS9: "Profit and Lace")
In 2375, Janel Tigan asked his sister Ezri Dax if he required a sonic shower before she would hug him. (DS9: "Prodigal Daughter")