Protagonist Villains is a trope found in any know media is about characters who are evil in every way, but are the main focus of their respective stories.
Why they are different form other one time characters or heroic protagonist?
A Protagonist Villain is, simply put, a character who is the central focus of his or her story yet does not have the traditional "heroic" qualities of a protagonist (they may cheat, steal or worse). These are the villains that are the protagonists of their own stories. Should they do have heroic qualities, they also considered as Anti-Hero.
Examples of this kind of character would be Pinky and Brain (who sought world-domination), Eric Cartman (many episodes focus on it when it comes to Jewish xenophobia), many GTA protagonists (who are criminals), and Horror characters such as Damien Thorn.
Some can be falsely seen as the "protagonist hero", as he/she is seen on the cover of the movie/book and there for the reader/watcher are being lured to believe that the character is the main hero of the story. One example is Major Chip Hazard from Small Soldiers.
Others of this kind can appear as harsh characters but mostly they are misguided Heroes, Girl Meme is a good examples of this, as she only wanted to find out the meaning of love and companion (Though she is actually an antagonist hero, but still count).
NOTE: Commonly Complete Monsters can never really be Added at this category, Although they are rare examples such as Light Yagami (Manga Universe Only), The Plutonian (From Iredeemable Comics), And Superboy-Prime (From DC Comics) Are Good Example of this.
This page is only for villains who are the Protagonist of their respective universes, in addition some of that can redeem at the end of his episode/installment, Girl Meme from Nihon Animator Mihonichi: I'm a Girl is a perfect example of this.