This villain trope is for the characters we love to hate.
"Love to hate" is a term used to describe villains we enjoy. We love them for various reasons, such as excellent character development, being hilarious, getting the best writing, having a memorable personality, or being lucky enough to have a top-notch performance by their actor. But at the same time, the fans still want to see the baddie get their comeuppance in the end. There is still hatred, after all. Also named Heel Hate.
Faux Affably Evil, Affably Evil, Laughably Evil, Magnificent Bastard, and Card-Carrying Villain are sub tropes for different reasons. Being a Large Ham definitely helps (see also Evil Is Hammy). An Anti-Villain is often one of these due to the character development they often get. A villain who's The Scrappy or Creator's Pet is typically not eligible for this trope as are some villains who cross the Moral Event Horizon. Jerkasses are also usually not eligible for this trope, but it's possible for them to be eligible if they're sufficiently entertaining enough.
Not to be confused with Draco in Leather Pants, which is when a villain's fandom will gloss over the actual nastiness of a villain and insist that they are just misunderstood, i.e they throw their sympathy to them even though they don't deserve it; in this, fans acknowledge the atrocities the villain has done but they like them for their villain traits. Also not to be confused with when the heroes would love to hate the villain, but just can't manage it.
This is someone whom the fans enjoy and even maybe respect because of their unapologetic evilness. Their shamelessness is cathartic to us, and they retain their awesomeness as time goes by without Villain Decay.
In order to be eligible, a villain must be generally well-regarded/well-liked by said fandom while the same fandom still acknowledges they're a villain. Being here means that said villain is part of the pantheon of great villains. Therefore, it is possible for a Complete Monster to be listed here (for example, the Joker, Pennywise, and the Red Skull).
Compare Hate Sink, where a character's intended role is to be hated by the audience. They can fall under this trope if viewers find them engaging despite being someone for them to root against. Also compare Jerkass Dissonance, where a character becomes beloved in spite of being so detestable. Can also overlap with Rooting for the Empire if the villain is so entertaining the audience actually WANTS them to win.
See also Evil Is Cool, a major cause of this sentiment.