It's quite good. I'd recommend it, but with the caveat that it's about a cold, reptilian boy who can "conquer" any girl at will. In other words, he makes them fall in love with him. He calls himself the "God of Conquest". The show does lots of interesting, funny things with this premise, but it's still something you might be nervous about letting normal people catch you watching. There's no way of describing it in one line without making it sound creepy.
THE HERO: Keima Katsuragi, game-player extraordinaire. He loves romance video games. He's the world's greatest player at them, but an utter failure in the real world. Characteristically he denies that that's any kind of flaw on his part, instead putting all the blame on defective reality. He calls reality a shoddy game. Real girls are a nightmare. They don't fit into a genre, they're self-willed and they call him a creepy no-life otaku (which he is). When the show begins, he's an abusive, arrogant jerk who won't stop playing his hand-held gaming console even when he's having a conversation with you.
In short, he's a virtual romance master who'd run a mile from the real kind. He seems to have no sex drive at all. (He never tries to take any of his conquests to the next stage, for instance, although I think I did spot him getting flustered by Kanon's cleavage.) He's appalled at having to woo real girls. He wouldn't even have cared about his target's gender, because that's irrelevant to the gaming challenge and he doesn't want a real relationship anyway.
THE SITUATION: Hell needs you! More specifically it needs Keima. There are lots of Loose Souls in the world, getting inside people and cracking open their personality flaws. The only way to save them is to fill the cracks in their souls... WITH LOVE! Keima isn't interested, so they give him an explosive collar. (Good news: driving out the Loose Souls will erase the girls' memories of falling in love with you. This neatly avoids the story problems that would have been created by an ever-growing harem. No memories, so no harem!)
THE SUPPORTING CAST: some of these are demons from Hell, but that's not as bad as it sounds. Hell's had a change of management and they're all quite nice down there, these days. Keima's main ally is Elsie, a childish idiot who's full of enthusiasm, but not very competent. She wears a scarf made of magical candyfloss and she's adorable. He's horrible to her. Beyond that, though, are of course the various girls.
The show's split into four episode blocks, each with its own distinct personality.
SEASON ONE (12 episodes, 2010) - this season is pretty much what you're imagining. Keima conquers some girls, most of whom have extreme personality issues. Mio Aoyama is proud to the point of self-delusion, Kanon can be needy and annoying and of course Shiori the librarian is even worse with other people than Keima is. He helps them all, although there are non-conquest episodes too.
It's a surprisingly good season. I'd have happily watched a whole show like this. One of its minor quirks is comedic callousness, so for instance Elsie's tactic for getting into Keima's household is horrifying (and funny), while the show's cheerful Keima-abuse will be cathartic to anyone who thinks he needs defenestrating. (He deserves it all and brings it on himself, obviously. Later seasons will give him character development.) That said, though, the show's also deconstructing lots of genre cliches, so for instance Tomoko agreed with all the brutal things Keima was saying about idols in ep.5.
SEASON TWO (12 episodes, 2011) - shakes up the formula. It's interesting. It invents new kinds of stories to tell in this universe and new ways of challenging Keima's assumptions. Keima isn't always the one who fills the target's emptiness. There's a realistic girl who foils all of Keima's tactics and breaks him as a human being, simply by not having neat characterisation. There's a teacher who identifies Keima as a problem student (and hilariously he can't work out why!), inverting the show's romance-driven patterns because she's the one targeting him.
OVAs (4 episodes, 2011-2013) - I don't think any of these have Keima conquering anyone. They're refreshing because they're Keima-light, instead focusing more on the supporting cast. I enjoyed these a lot and I'd particularly recommend the fourth OVA episode, which is lovely. (Kanon invites a young fan on to her magical girl TV show, Weather Girl Magical Girl Kanon.) Warning: these OVAs are on the DVDs, but I think they might not be on Crunchyroll. I enjoyed them and I'd recommend them, but you'll understand everything just fine if you go straight from Season 2 to Season 3. They're not essential.
SEASON THREE (12 episodes, 2013) - breaks the show's formula in two and invents a new one. I'm torn. On the one hand, this season is clearly the one with the most meaningful storyline, the strongest story arc and the deepest character development. People learn things and are changed, permanently. That includes Keima. On the other hand, though, this season also has Keima having to woo multiple girls simultaneously, which I personally found a bit uncomfortable to watch. At times, I didn't like it. It can be funny, though, and the show has impeccable reasons for everything. Occasionally I'd get exasperated at Keima keeping the truth from the girls even when just telling everything looked like the natural and correct action, but this eventually got addressed. (Keima's arrogant enough to think that he can manage any situation that's familiar from his games, whereas explaining about hell, heaven, demons and goddesses strikes him as an unmanageable risk.)
SEASON FOUR (manga-only) - I haven't read it, but I know it exists. The anime has quite a striking and slightly open-ended finale. I think it's very good, but those of you who want to know what eventually happens will have the option of reading up online about the manga's unadapted final story arc.
It's a pretty good show, I think. It's certainly far better and more interesting than you'd expect from its premise. It's also funny! On the downside, the art style doesn't always make it easy to remember which girl's which, especially when someone from Season 1 becomes a background supporting character in Season 2. There's also the fundamental improbability of Keima becoming such a babe magnet. It's worth a spin, though, if you're comfortable with what it is.
