Glossary Entry : Harem
Abstract:   This is partly a genre, partly a formula and partly a recognition
of the young male anime audience. The important element is that one
male, generally shy and reserved around females, finds himself
suddenly at the center of a story involving multiple females who
have a tendency towards being extremely cute and sexy. It's not
exactly the most highbrow form of anime.
However it's not quite as stupid as some people make it out to be. At least not always. Certainly the shoujo crowd tend to take this genre as evidence of how sexist and simplistic shounen anime can be. It would be nice if they didn't have quite so many examples to support them in their arguments. Many of these shows do feature a young male suddenly inheriting a bevy of beautiful and busty females who, for various reasons, are eager to serve him.
However many of these shows are not simply teenage wish fulfilment. They are also a very convenient structure to showcase a number of very strong, and highly independent, characters who would not interact in a stable fashion without some, largely inert, core for them to orbit around. In other words these characters bicker, compete and interact with the lead who is a passive or uncommitted centre point for the show. If the lead were removed, or a strong character, the lack of stability would cause the cast to fragment. This is why Tenchi, the archetypal example of this genre, would cause the show to end should he make a declaration of love to one of the females who compete for his attention (which is a subset significantly less than the whole cast (#1)).
As such this format allows an exciting cast of characters, who interact fiercely with each other, to be packed into a very tight space where they are forced to interact frequently. And being a stable cast they can also be involved in episodic adventures or `wobbles' in the various orbits. However I would be lying if I did not admit that, in most cases (as in anime in general) these strong characters tend to be attractive females. This also allows for `romantic tension' and `jealousy' which is a relatively easy way to power plots.
One possible way to tell the two apart is by the power relationship between the males and females. If the females are maids, servants or eager to please then it could readily be called male fantasy. If the females are likely to thump, manipulate or threaten the lead then it may be that it is the females who are actually the stars of the show (albeit for a male audience).
It may be worth noting that in both cases the male tends to be inert for some reason. This is partly to maintain the stability of the shows characters. However it may also be to allow the viewer to place himself in the character at the center. Thus a weak and shy male at the center of many females may allow the male viewer to perceive himself being rather more direct in the same situation. Of course there is also the factor that if the male lead in a harem show did fully exploit his situation we'd be firmly in Hentai territory.
#1 -- At least in most normal people's opinion. Tenchi Muyo fans seem to enjoy widening the net by opening his possible romantic options to just about any female in the show.