Harem itself has 3 subgenres:
Stories where the lead has competing love interests aka shipping wars. This is probably the most common type, where the lead is popular for seemingly plot convenient reasons that make little to no sense. The story is about who the lead will choose to end with shennaningans/situations to appeal to the audience to root for the best girl/wifey/partner. In most cases, the visual appeal is what matters most rather than the progression or character development. 5toubun no Hanayome (5 identical sisters for their tutor) or Rokudou no Onna-tachi (MC has a magic that makes deliquent girls fall for him) are good examples of this categorization
Stories where the lead takes everyone/most: The sub genre that is best representative of what harem should be in my opinion. Whether its plot mechanic to take multiple partners or because the MC acts on their desire to take/woo the heart of whomever they are pursuing, the story ends with the MC in a romantic relationship with multiple girls. In some cases, even has started a proper polygamy and/or family with the leads. Sekirei or DearS are good examples of this genre
Stories that aren't really harem: There are quite a few stories that are attributed as harem, but really aren't. Whether its the story of an MC that is in short term relationship, breaks up and gets into a new relationship (aka continual rebounder) or the MC has an affair (NTR) on their partner with another person, they are labeled harem but really aren't. Beyond Myriad Peoples is a good example (all the MC does is talk with women on simple conversational topics and while he does have a couple of women express fleeting interest once in a while, nothing happens to indicate a harem, unless having a huge fanclub worshipping you is supposed to be harem)