A bark was the sound made by various animals, such as Earth dogs, as a "warning" or "attention" vocalization; it was often used by humanoids as an onomatopoeia.
Jonathan Archer's Beagle Porthos would often bark in order to indicate he was hungry, especially for cheese. (ENT: "Cold Front", "Singularity") Porthos also barked and growled whenever he sensed something out of the usual, such as an invisible Suliban in their quarters. (ENT: "Cold Front", "Vox Sola", "Doctor's Orders")
Keyla's pet Rhylo barked at Archer and Porthos, upon watching them arrive at their villa on Risa. Later Porthos barked then growled at Rhylo, who had wandered onto Archer's deck. (ENT: "Two Days and Two Nights")
After the Ferengi Grish was unable to lock on to Porthos bark (aka "its language"), his partner Muk chided him for being a fool, as the dog was a lower lifeform, and "probably the captain's next meal." (ENT: "Acquisition")
In 2369, during an argument with Beverly Crusher, Q transformed her into a dog that barked several times before Amanda Rogers returned her to Human form. (TNG: "True Q")
The proverb, which essentially stated "his bark is worse than his bite", meant someone seemed more hostile than was the case.
Trevis, Neelix, and Naomi Wildman all made "silly puns" about the tree character; twice about his bark being worse than his bite, and once about getting hot under the bark. (VOY: "Once Upon a Time")
The Krenim commandant encountered by USS Voyager warned Captain Kathryn Janeway that where they would be destroyed if they entered Krenim space, and that there would be "no discussion!" or "no compromise!" before retreat. Commander Chakotay noted that "his bark's obviously worse than his bite;" while Janeway observed that "he seems rather intent." (VOY: "Year of Hell")