A slingshot was a hand-held weapon used to propel projectiles across a short distance.
David Rabin used a slingshot to hunt food and to defend himself and Spock while traveling across Vulcan's Forge. He noted that his namesake, the legendary King David of Israel, was also expert in his use of the slingshot, and hoped this was a good omen as they faced an adversary as formidable as Goliath. (TOS novel: Vulcan's Forge)
During their quest for the legendary "creature of the mountain" on Theata Leonis, Kirk, Spock, McCoy and the Atyansa boy Meresh used forked staves as slingshots to fight the monsters guarding the mountain and to bring down rocks to delay a group of Klingons from entering a cave. (TOS comic: "Epic Proportions")
"Slingshot" could also refer to the slingshot effect, which uses the gravity of a star to snap a ship through time, like a slingshot. (TOS episodes: "Tomorrow is Yesterday", "Assignment: Earth", TOS movie: The Voyage Home, Star Trek novel: Engines of Destiny)