The first bullet was made by Smith and Wesson. They patented the 22 rim fire cartridge on August 8th, 1860. Originally, bullets were made of metal, stone, or purpose-made clay balls. But before Smith and Wesson made the 22 rim fire cartridge, bullets were made as sling ammunition for hunting and other various activities. 

These are lead sling bullets from Athens, in 4th century BC. 

Finally, when firearms were developed these stone or metal bullets were placed in front of a propellant charge of gunpowder at the end of a closed tube. From 1500 to 1800 bullets changed very little. They remained small spherical lead "rounds," only changing slightly in diameter.