This is a modern cartridge: 

1.  the bullet/projectile

2.  the case; holds all parts together

3.  the propellant (ie: gunpowder)

4.  the rim; a place to grip the casing to remove it from the chamber once fired

5.  the primer; ignites the propellant

     Between 1854 and 1857 Sir Joseph Whitworth conducted a series of rifle experiments and proved the advantages of an elongated bullet. In 1882 Major Eduard Rubin invented the copper jacketed bullet - and elongated bullet with a lead core in a copper jacket. The surface of lead bullets fired at a high velocity may melt due to hot gases behind the bullet. Because copper has a higher melting point and hardness, copper jacketed bullets allow greater muzzle velocities. 

The word "bullet" is derived from the French word "boulette" which basically means little ball.  The original musket bullet was a spherical lead ball.

Musket Ball

These musket balls were no more than one inch in diameter. 

Among the first pointed/"conical" bullets were those designed by Captain John Norton of the British army in 1823. His bullet had a hollow base. The British Board of Ordinance rejected it because spherical bullets had been in use for 300 years. Then William Greener, an English gunsmith, invented the Greener bullet in 1836. It was very similar to Norton's design, except that the hollow base was fitted with a wooden plug. Unfortunately it too was rejected for military use because, being two parts, it would be too complicated to produce. In 1847, Claude-Etienne Minie introduced the soft lead minie ball. This too was similar to the Greener bullet. In the American Civil War roughly 90% of casualties were caused by the minie ball.