A full metal jacket (or FMJ) is a bullet encased in a copper alloy such as gilding metal, or cupronickel, or a steel alloy shell. This shell can extend around all of the bullet or often just the front and sides with the rear left as exposed lead. The jacket allows for higher muzzle velocities than bare lead without depositing significant amounts of metal in the bore.
It also prevents damage to bores from steel or armor piercing core materials. This bullet type distinguishes itself from hollow point bullets. The symbolic meaning of the term is that the recruits on Parris Island have to go through a rigorous training program to turn them into soldiers capable of following orders without question, surviving in combat and to kill their enemies without having any moral complications about doing so.
Also, Joker's time in combat in Vietnam is indicative of this concept: by the end of the film, he is a hardened soldier without fear. If you look at the thematic and symbolic meaning of a full metal jacketed bullet, the recruits have developed a hardened exterior for themselves.