I'll have a crack of combining all the info in 1 post....
Copper Jacketed as Oldbloke points out consists of a copper cup punch from copper alloy sheet into which a lead slug is inserted before being shaped with a series of dies (swaging) They originally developed jacketed bullets to overcome the limitations of velocity with lead. A lead bullets fouls rifles barrels and increases fouling exponentially with velocity meaning that as velocity increased barrels had to be cleaned more often which was not an ideal situation to be in when trying to shoot the enemy or practical for use in machine guns.
https://www.youtube.com/watch?v=Hkr1NS7bEUUCopper Plated / Wiped / Coated (same thing) is generally done via electrolysis to deposit a very thin layer of copper. The reason behind this is mainly for lubrication and it's just thick enough to allowing slightly higher velocities than plain uncoated lead. Lead needs to be lubricated to reduce fouling, older style lubes were a wax/grease type but that isn't conducive to mass production. Newer style coatings using polymers or 2 pak style paints are popular but require multiple applications and drying time between coats so the next big thing was copper plating as it was a single application and thousands of projectiles could be plated very quickly in one application with minimal drying time.
Cast lead / Hard Cast lead - Lead hardness can be measured in Brinell units (HB) and depending on the use of the projectile you may want a slightly harder projectile for use in say a rifled barrel to reduce lead fouling where as in a smooth bore gun you can use a softer lead. Hardness can be controlled by "Alloying", mixing lead with small amounts of Antimony and or tin to achieve the desired Brinell hardness. I think Noisydad does a lot of casting and maybe he can add more info if required.