Anime: Chobits
Manga: Chobits, Tsubasa: RESERVoir CHRoNiCLE
Games: Chobits (GBA)
Spoilers are coloured.
In Chobits, humanoid robots known as persocons (also the Japanese word for PC) are the new big thing, even though they are stupidly expensive. Chii was found abandoned by the protagonist, Hideki, and he took her in, losing her old memory disc in the process. After restarting her (and what a place to put the start switch), she can only say 'chii', and so Hideki names her accordingly. It soon becomes apparent that Chii is a custom model with a powerful protection program installed, and there's a possibility that she is a 'Chobit', a legendary persocon with the ability to develop emotions.
Hideki has to teach Chii everything again, and so Chii sees the world as through the eyes of a child, being naive and generally innocent. Naturally, over the course of the series she falls for Hideki, but can he return her feelings, knowing she's a persocon? And who is the Dark Chii that she keeps seeing?
It turns out that Dark Chii is Freya, Chii's sister (back when Chii was known as Elda), who fell in love with their creator and destroyed herself when he couldn't return her feelings. As a result their creator added a special program to Chii that would affect all persocons (for better or worse?) if activated. It could activate depending on Chii's most important person's feelings for her.
The nature of the program is never actually explained in the anime, and I'm not even sure if it activates or not, although I'm inclined to think it does, and therefore has a different effect to the program in the manga. In the manga, the program would destroy all persocons' person recognition abilities and thus render it impossible for them to fall in love and be rejected. Needless to say, it doesn't run, since Hideki loves Chii.
And so, is Chii a Chobit? Yes, but she does not have emotions or anything; that was just a false rumour. Chobit was just a name the creator used for his two 'children', and no persocon can function outside of its programming.
Chii's creator was none other than Ichiro Mihara, or Icchan, from Angelic Layer, although he's never actually named in the anime.The Angelic Layer and Chobits mangas are related, but the animes are not. By the time of the Chobits manga, the Angelic Layer game has dropped in popularity and pretty much no longer exists. Whereas I'm not so keen on the Angelic Layer manga (love the anime though), I feel the opposite about Chobits, and like the manga, but dislike the anime (although it does pick up in the second half).
Chii has a much smaller role in Tsubasa, at least so far. In an alternative universe, she is a magical creation of Fye, but he had to remove her as he prepared to flee his universe. It is possible that she will return (or they could encounter a different Chii).