Harem is a genre/plot frame that is mostly present in the Japanese media of Anime, Manga, Light Novels, Visual Novels, and Video Games.
It normally takes place in a High School setting, with one male lead, and at least three, often a lot more, girls who are romantically interested in him. (Gender Inverted Examples also exist). Usually, each girl personifies a single classic characterization archetype.
The protagonist either takes it as an Unwanted Harem, or reacts as a Harem Seeker, or Oblivious to Love.
Works where a Harem Genre set-up is put in front of other relationship dynamics, and the plot is written with the intention to keep the "race" for the male protagonist's heart as tied as possible, are known as the Balanced Harem subgenre.
In other cases, when a work uses this genre together with a more traditional Romantic ComedyOfficial Couple relationship, that is called the Supporting Harem subgenre.
Many of these works are also shared with the Ecchi genre, as it is harder to find a Harem story that isn't filled with gratuitous Fanservice than one that is.
Contrast Royal Harem, which is about the literal, traditional sort of harem as found in a royal palace.
Sakura Taisen, naturally, as it's based on a game that is partially a Dating Sim. Therefore we have Sakura, Maria, Kouran, Sumire, Kanna and Orihime, and Leni and Iris are too young but seem to be net in line.
Code Geass: It seems like everyone in the series is in love with Lelouch. In the first season, Lelouch/Zero accumulates a rather large group of girls who have feelings for him, including C.C., Kallen, Shirley, Milly, Kaguya, and even his half-sister Euphemia. He just doesn't do anything about it, since for the most part he's too busy trying to overthrow the government to pursue romance. The staff has said that the second season will deal more with his romantic entanglements (and note that all this is ignoring the Brother-Sister Incest and Ho Yay-slash-Foe Yay ships with Nunnally and Suzaku, respectively, and a combination of both those tropes (though a downplayed one that doesn't get as much focus) with Schneizel, who refers to Lelouch as the "one he loved and feared the most").
The second season seems to poke fun at this; as a result of his Ninja Maid running around in a Lelouch disguise and being generally nicer, Lelouch ends up scheduled to go on dates with no less than 108 girls. And there's Kaguya fantasizing about polyamory...
And yes, the staff sort-of delivered it. Not only Rolo became a secondGay Option (who was all "Bitch, PLEASE!" at Sayoko for the 108 dates), but there is more focus on Lelouch's love life... Specially when it hit Fatal Attractor levels and later Suzaku manages to ensnare the position of The Dragon to Lelouch, with all the Ho Yay squeals it implies. See the entry.
This was played with in one of the first season's audio dramas where Shirley considers asking him to go to a concert with her and imagines him revealing that he was in love with and getting married to Kallen, Nunnally, Suzaku, Milly, Rivalz, and even Arthur the cat in that order. At the end they all decide to share him and invite Shirley in on the action. Note that Nina was not in the fantasy and it was set only hours before the Table-Kun incident.
Urusei Yatsura shows how this was already a cliche back in the 1980s — Ataru Moroboshi, who is actively trying to assemble a harem, repulses most girls and has to "settle" for the alien princess Lum, who lives in his closet and thinks they're married.
A particularly interesting (and unusually non-romantic) variation on this trope can be found in the anime Sister Princess, in which a young man discovers that he has a baker's dozen half-sisters he never knew existed. Having grown up alone except for a few servants, he finds their slavish adoration of their "big brother" somewhat overwhelming, but grows to love and treasure them all.
Infinite Stratos: Slight variance on it in that there's some Powered Armor that only girls can use, but for some reason, one guy is able to use it as well, and they promptly put him in the all girls school to learn how to use those things. Naturally, being the only guy in an all girl school...
Kampfer: Here the main guy turns into a girl thanks to a magical MacGuffin. Who's probably hotter than the rest of the main girls.
Saber Marionette J, in which the entire harem is made up of Robot Girls and one guy (on an all-male world, where this would be expected). In its various sequels, the one real girl in the entire world also apparently wants in on the harem.
The manga Ai Kora (from the creator of Midori Days) plays with this: Hachibe Maeda, a guy with very particular tastes in women definitely wants the harem so long as they have exactly one of the attributes he's looking for in a girl: Tsubame-sensei, the Harem Nanny, has long, streamlined legs; Sakurako, the Tsundere, has big, blue eyes; Yukari, the Meganekko, has big, perky breasts like a bullet train; and Kirino, the Little Miss Snarker, has a husky voice. Later we meet Ayame, the Ojou, who has Maeda's ideal waist. This is taken to the extreme with Haiji, who has the ideal ass but is a guy. And a bisexual, no less.
Rune Soldier Louie looks like a Harem Anime at a glance — Louie does get unexpectedly involved with a bunch of beautiful women, after all — but it never really plays up any of the genre's tropes beyond Louie being stuck between Ila and Melissa (and Melissa doesn't even like him that much); Merill and Jeannie just treat him as a cool brotherly type/drinking partner. In any case, the action and comedy take precedence over the relationships.
According to an un-canon pre-anime sketch, this happens in Higurashi: When They Cry. After Keiichi goes even more bishonen than he already is, it's shown that Rena, Mion, Rika and Satoko all like him.
It's a Drama CD, actually. Someone just drew pictures for it, and again, non-cannon. On the other hand, we have one of the Kira episodes that's a better example, with Shion, Rena, and Mion all out for Keiichi.
Nadja Applefield (or better said, Nadja Preminger), from Ashita no Nadja is another of the female examples, having one of these (with four pre-teen boys and three guys in their mid-to-late teens) when she's just thirteen.
Can't forget the lesbian Love Triangle in Kashimashi: Girl Meets Girl from which Hazumu has to choose either Yasuna or Tomari. In addition to that loads of other characters be they guys, girls, aliens, or family members all have a thing for Hazumu.
Played with in the manga Otomari Honey, where we see the situation from the perspective of one of the haremettes - she goes to the house of a boy she has a crush on to ask if she can live with him to get away from her own home, only to find that he's already got three other girls with the same story living in his room.
Ichigo 100%: Manaka has a rather large group of different girls after him in Aya, Satsuki, Tsukasa, Kozue, and Yui
Tomoki Sakurai from Heavens Lost Property has a harem consisting of Ikaros (who still hasn't quite realised it), Nymph, Sohara, Hiyori, and possibly Astraea. However, he's way too busy thinking of perverted schemes and being a pervert to actually notice. Only Hiyori has confessed to him, and he's even stunned upon receiving her confession.
Hayate the Combat Butler, a strange combination of the guy being even more clueless than usual and the girls mostly being knowledgeable about and completely alright with competing for his affection between them. Several of them actually work very hard to get one of the other girls to win!
RPG Shooter Starwish is a hybrid RPG shooter where the harem consists of 5 girls (only 4 are available at a time) and 2 guys, with a special ending if you can get all of the individual endings (and another one - featuring a fifth girl who was not previously in the story except for a few lines at the beginning, which the protagonist assumed was All Just a Dream - if you don't).
RPG Maker 2000 game Romancing Walker features a pretty boy hero and his six love interests.
In Bully, the main character amasses himself a nice harem.
In Scarface: The World is Yours Tony Montana can obtain a large collection of trophy girlfriends.
Agarest Senki requires you to pick one girl out of three in each generation after you've spent some time with them. The system works similarly to Phantasy Star III's syetem. Rex, the final hero, however, can end up with one of any girl you've gotten in your party so far of the maximum number of nine in the Golden Ending.
Any game created by Bioware has romance side-plots reguarding the player character. Most of the time it's just a love triangle, but the ammount of people your male/female Shepard can become interested by someone in Mass Effect 2 pretty much borderlines a harem.
Hakuoro of Utawarerumono sleeps with six women throughout the game, five of whom love him. Kuuya, empress of the bunnypeople, also seems to love him and offers her best friend as a concubine without knowing quite what that means. Mikoto and possibly Mutsumi in the back story. Yet he's entirely oblivious of how most of them feel until the very end of the game, thinking of them as mostly friends/retainers and of the sex as being rather casual.
Considering that he's the king, who also comes from modern society, where sex IS casual, and they live in a world which still has EMPERORS... it makes sense.
Many visual novel plots revolve around especially in eroge, but this trope isn't eroge exclusive.
Tsukihime. Five girls with their own routes plus one without, having been Demoted to Extra. Most of them are quite violent about it too; Akiha flatly hates Ciel (who hates her back) and Arcueid (who doesn't seem to know what hate is) and those two have a 'weird friendship' that occasionally includes trying to kill each other. Akira is added in the first fandisc, Kagetsu Tohya seems to add Len considering she expresses a preference for having sex with him when making a contract and implies most of his classmates are attracted. Melty Blood adds Sion.
Higurashi: When They Cry initially seems as though it'll be one of these, what with the main character, Keiichi, being in a Games Club consisting of only four girls and himself (later adding two more girls, Shion Sonozaki and Hanyuu), and the first parts of each arc focuses on their shenanigans. Soon, though, it's revealed that this series is about somethingelseentirely. Not to mention that in the second half of the series the position of main character shifts to Rika instead of Keiichi. It's also worth noting that only Mion and possibly Rena are actually hinted to be in love with Keiichi in the series proper, since Shion is in love with someone else, Satoko and Rika are too young, and Hanyuu is a Physical God who was married and had a daughter when she was still human. In the end, none of the girls end up with Keiichi, since the series is ultimately not about romance but The Power of Friendship.
Key Visual Arts' AIR, Kanon, and Clannad feature an ordinary (or not so ordinary) high school student who attracts girls of every moe and general description type into a long-running harem.