The term harem anime refers to an anime series with a main character who has some kind of romantic connection with two or more other characters. Usually, the main character is male and the other characters are female. The reverse does exist, as do same-sex harems, but they aren't as common. The genre has significant crossover with ecchi anime, since it tends to involve extreme levels of fan service.
One of the biggest problems with harem anime, aside from the blatant objectification of women that's endemic to the genre, is the fact that some of the protagonists are eye-rollingly boring. No, really, they're some of the worst anime protagonists out there, and they make for some of the worst anime boyfriends. Kirito is nobody's favorite character in Sword Art Online, and if you remember Love Hina at all, you definitely don't remember it for Keitaro.
This might be because the intended audiences for most harem anime is straight men and teenage boys projecting themselves onto the male main character. You know, like how teenage girls did with the oh-so-bland Bella from Twilight? Projection is a heck of a lot easier to accomplish when the protagonist is a blank slate.
This doesn't exactly make for great storytelling. While some viewers are happy to use protagonists as wish fulfillment, others want, you know, actual character development. Or, at minimum, non-horrible anime relationships to project onto. Sadly, many of these anime heroes are so mind-numbingly dull, they shouldn't have harems at all.
Considering the fact that School Days ends with Makoto's harem straight-up murdering him, you'd think there'd be something about him to make him worthy of that kind of passion. There is not. Makoto's main personality trait is his propensity for cheating on (and manipulating) women.
The dude makes out with a girl in front of another girl (who he supposedly got pregnant), and he has sex with multiple girls concurrently despite their clear indication they did not sign on for polyamory. While this does make Makoto slightly... unique in terms of harem protagonists, there's not much to him besides general douchebaggery. Weirdly, the viewer is expected to feel sorry for him when he gets killed. Empathy does have a limit, School Days.
After failing his college entrance exams, Keitaro Urashima becomes the manager of his grandmother's women-only apartment complex. There, he meets a group of women and proceeds to be the least professional building manager in history. They all have some degree of crush on him, and he regularly trips and lands face-first in their cleavage.
He ultimately marries the one whose tsundere escapades border on domestic abuse, Naru Narusegawa. Keitaro marries her because his only character trait is his desire to track down his childhood bestie, who happens to be Naru. The fact that she constantly punches him in the face is apparently just whatever.
Created By: Ken Akamatsu
Yuto Amakawa From Omamori Himari
Yuto Amakawa is an orphan whose parents left him a magical relic prior to their deaths. His (weirdly same-aged) childhood friend has been taking care of him since he was orphaned. As it turns out, the relic is a link to his genetic destiny as a demon slayer. He ends up surrounded by large-breasted, big-eyed demon girls who are all madly in love with him.
So, what unique personal attributes does Yuto have to inspire such ardor? One. He's allergic to cats. It makes his cohabitation with a cat demon mildly problematic. That's it. When the only trait that makes you any different from any other person on the planet can be tamped down with a couple of Claritin, you know you're doing something wrong.
Rather than actually having his own harem, Ataru just desperately wants one. The only thing Ataru has resembling a relationship is with the alien princess Lum, who is so madly in love with him that she electrocutes him whenever he looks at other girls. Despite this, Ataru keeps trying to collect harem members, joyfully hitting on anyone with a vagina and a pulse. It's a miracle he doesn't die by the end of the series, and it's an enduring mystery why Lum loves him so much. The fact that he doesn't get the harem he's striving for, however, isn't mysterious at all.
Created By: Rumiko Takahashi