In the PPC, assassination is the name given to the act and art of killing a Mary Sue. Assassination is not the same thing as murder, and therefore it is not right to refer to a PPC assassin as a murderer. In fact, doing so is insulting. PPC assassins don't kill real people, for one thing; and anyway, common murderers have no standards and no style at all.
PPC agents typically assassinate Sues in a lavish and painful manner, usually with the Sue restrained in some capacity, but that isn't always the case, and outright torture is frowned upon. Some agents kill the target mundanely (for example, by gunshot or sword wound), and occasionally the Sue fights back (in which case some agents have been wounded or even killed). Assassination methods vary from mission to mission.
Effects of Assassination on the Continuum
The presence of a Sue distorts the continuum. In badfic with a single, extreme Mary Sue as the main problem, assassination of the Sue will cause the canon to reassert itself instantly; in other cases, canon characters must be neuralyzed, OCs removed, or even entire countries excised. Once all major distortions are removed, the canon snaps back to normal in anywhere from a split second to a few minutes.