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 skill. For every kill that the weapon makes afterward, its chances of landing a critical hit improve. The bonus caps out on 100 kills or 50% bonus.
TearRing Saga uses the same bonus as in Genealogy of the Holy War, with the exception of redundancy of Critical skill. There are also three weapons that start with certain amount of kills (Ch.1 Iron Sword and Ch. 3 Iron Lance with 50 kills and Ch. 4 Hand Bow with 80 kills)