This potentially deadly weapon -- not a toy! -- is known as a sling or shepherd's sling:
People have probably been slinging rocks, clay balls, darts, and other blunt or pointy objects -- at animals and at each other -- since Neolithic times.
The bullet-shaped lead projectiles shown below are in fact lead sling bullets from Athens, 4th century BC, with "a winged thunderbolt engraved on one side and the inscription “take that” on the other side." -- Wikipedia
According to the Roman military historian Vegetius,
The inhabitants of the Balearic Islands are said to have been the inventors of slings, and to have managed them with surprising dexterity, owing to the manner of bringing up their children. The children were not allowed to have their food by their mothers till they had first struck it with their sling. Soldiers, notwithstanding their defensive armour, are often more annoyed by the round stones from the sling than by all the arrows of the enemy. Stones kill without mangling the body, and the contusion is mortal without loss of blood. It is universally known the ancients employed slingers in all their engagements. There is the greater reason for instructing all troops, without exception, in this exercise, as the sling cannot be reckoned any encumbrance, and often is of the greatest service, especially when they are obliged to engage in stony places, to defend a mountain or an eminence, or to repulse an enemy at the attack of a castle or city.