Since there is no way to directly attack another player outside of a Ballista match, people who want to kill other player characters (PK = Player Kill) have to resort to other means. The way they do this in FFXI is by getting a monster to kill the player on their behalf (MPK = Monster Player Kill).
This is usually done by intentionally pulling one or more monsters that aggro near the intended victim. The puller then does something to lose hate (Warp, Hide, etc), and allows the mob to aggro to and hopefully kill the victim.
This is against the Terms of Service agreement. MPK'ing was explicitly added recently as a forbidden act, and can result in a suspention of your account, character deletion, or bans.
I only described how it is done so that you can recognize it when/if it is ever done to you. Keep in mind that sometimes accidents do happen, and people will unintentionally die while training an aggro monster, or through sheer stupidity may purposely come near you and ask for your help, only to die and have the mob engage you. The ones to watch out for and report to a GM are instances where the person doing it does so with the sole intention of killing you or your party, or interfering with your gameplay.