The Weapon Kill bonus refers to the process of receiving double weapon experience when a weapon kills an enemy. This bonus was implemented in all three of the GBA games and vastly sped up the process of raising weapon ranks, but was subsequently removed in Fire Emblem: Path of Radiance, explaining why it is difficult to raise weapon ranks in that game.
Fire Emblem: Genealogy of the Holy War has an alternate Weapon Kill bonus. Once a character's weapon has slain 50 enemies, it gains the Critical ability. For every kill that the weapon makes afterward, its chances of landing a critical hit improve.