The first repeating projectile weapon is probably the Chu Ko Nu, the repeating crossbow invented in China (often misattributed to legendary strategist Zhuge Lian) around the 4th century BC. This was a light crossbow with a hopper full of bolts and a lever that the operator could move back and forth to repeatedly draw the bow and fire rapidly. It does not meet the criteria of 1 shot per second, but deserves an honourable mention.

By the late 18th century, a number of repeating firearms would emerge. As recoil driven actions were not yet invented, these usually relied on having either multiple barrels or multiple locks (things that lit the gunpowder). The multiple lock configuration is known as superposed loads; the user would load powder-ball-powder-ball, and the locks would be positioned such that you could fire each in sequence. Naturally, this was not a great design as fouling, failure to fire and operator error could lead to the rounds being fired out of sequence and, in scientific terms, kablooey.

Multiple barrel guns were more successful in a military context. the Nock gun, designed in 1779, was a massive seven-barreled monstrosity that fired all seven barrels at the same time for ship-to-ship combat. Numerous rifles and pepperboxes would follow a similar design. They technically fulfill your criteria of one shot per second, but isn't particularly repeating, since reloading between shots took quite a lot of time.

The first, really undisputable repeating firearms were known as volley guns, the first two being the French Mitrailleuse (prototype 1951, adopted 1866), which used multiple barrels that were fired in sequence, and the American Gatling gun (invented 1961, adopted 1862), which uses a cluster of barrels that were fed from a hopper and loaded and fired continuously using a rotating crank mechanism. These two, depending on whether you go buy adoption or invention, were the first repeating firearms.

Another honourable mention should go to the Girardoni Air Rifle (1779), a compressed air weapon that shot bullets using pressure from an air tank that was good for around 30 rounds. The air rifle was used briefly by the Austrian empire, and famously by Lewis and Clark as a demonstration of European technology on their expedition. It had relatively low power and refilling the air tank took a lot of time with the hand pump, and was a dead end in military technology.