Yui is a girl in Rito's class who seems more strict than the others and seems initially uptight about school morals. This attitude can be traced back to her childhood, given that the boys in her class would sexually harass her classmates by flipping their skirts, and picking on her friends. Yui took it upon herself to stand up for her classmates and came to detest boys and shamelessness as a result.
She quickly comes at odds with Lala and her free-spirited personality, which she sees as a hindrance towards making the school a more peaceful and better place. Likewise, she has a certain distaste for Rito, given that his frequent mishaps lead to misunderstandings concerning his true nature. At first, she believes him to be just another shameless pervert, but gradually she begins understanding this is not the case.
Eventually, she loosens up a bit, but she's generally more on her guard than the other girls. She becomes a part of Rito's harem all the same.
In the anime version it is also implied (albeit rather briefly) that she might have feelings for Rito. In the original manga version, however, this is made evident, to the point she even fantasizes about marrying Rito and adopting Celine as her own child. This also causes her attitude to shift, especially in the latter half of the manga, where she mellows out a lot, especially around Rito.
It should be noted that there is a huge rift in terms of how her personality is portrayed between the anime and the manga, especially in the first season where Yui is depicted as far stricter and harsher than she really is. Her introduction episode in the anime makes a very radical departure to her portrayal in the manga, depicting her as extremely puritanical and authoritarian. This is eventually toned down but this initial portrayal has also caused such a reputation to stick among anime-only viewers.
Likewise, a good deal of her scenes from the manga concerning her character development have either remained unadapted or given to Haruna instead. For example, in the OVAs it's Rito and Haruna who travel together to the fantasy RPG crafted by Lala's sisters, however, in the source material this role was originally given to Yui. It is uncertain why the directors chose to diminish her role in the series, given her popularity among the fanbase.
In the post-manga interview the authors Saki Hasemi and Kentaro Yabuki declared she was introduced as the only character meant to develop, in contrast to the rest of the cast which were written to sort of "reset" with each chapter, given the slice-of-life setting. This changed in To Love Ru Darkness, which was more story-driven, were most characters are given some sort of development.
