Revivable Pokemon are Pokemon identified as being somewhat/completely exempt from the "fainting=death" rule of the original ruleset.
"[identified Pokemon] can be "brought back to life" once/multiple times/indefinitely."
As one of the intentions of Nuzlockes is to get players attached to their Pokemon, wish to keep a particular Pokemon alive even if they faint at least once. Often the death of said Pokemon would severely hamper the plot a player has created in their documenting comic/story.
Many Nuzlockers feel that reviving Pokemon, for any reason, goes against the spirit of Nuzlockes, especially if it's used as a means of not admitting mistakes. As such, runs that identify revivable Pokemon are not considered true Nuzlocke Challenges (they are not allowed into the FCR section of the official Nuzlocke forums, instead being demoted to OA).
That said, many other Nuzlockers value strong plot reasons for naming revivable Pokemon, as such viewing use of this clause acceptable - but only if the clause is not abused. Examples of abuse include using an infinite (or a large finite) number of revives, naming an infinite (or a large finite) number of Pokemon this applies to, or identifying a Pokemon as revivable well after the comic/story was started.