Expanding bullets are intentionally designed to expand when they strike the target so that the diameter of the wound or the part affected by the bullet is increased while the penetration is reduced. Though earliest conventional bullets made of round lead lead balls also expanded to some extent. these were not designed specifically for this. However with development of guns with much more powerful explosives, it became necessary to use elongated bullets of smaller diameter. It also led to use of bullets made of harder materials other than lead. Such bullets did no expand much. As a result it became necessary to incorporate features in bullets that caused considerably higher expansion. Several different methods are employed to make expanding bullets.
A normal bullet passes through the target it is fired at with very less energy actually spent in causing damage to the target. These bullets have a thick layer made of brass or copper which keeps the bullet intact when it strikes the target. An expanding bullet on the other hand is designed to cause a lot more damage. Here the outer cover is quite weak or missing. So when the bullet is fired, on hitting the target it penetrates a little and by then the lead which is filled in the bullet comes out forming a mushroom shaped structure. Expanding bullets use more of the energy they have to cause damage to the target.
The Hague convention forbids the use of expanding bullets and makes their use a war crime, but in the US expanding bullets are the preferred ammunition and are used by the police and civilians alike.