Yandere is a character type in Japanese fiction who is genuinely in love with the object of their affections, but this love goes too far and drives them to brutal, pyschotic, or deranged behavior. The word is a portmanteau of the Japanese words yanderu (病んでる), which means "to be (mentally) ill" and deredere (デレデレ) or "lovey-dovey," and was coined by analogy with "tsundere" (another character type of someone who is initially cold but later grows warm and affectionate).
Unlike the more well-known tsundere trope, which always incorporates a shift in behavior over time, a yandere character's deranged or psychotic behavior can either be present from the start or can slowly develop over time. Classic examples of yandere behavior might include kidnapping one's crush, torturing the loved one's enemies, poisoning romantic rivals, or even murdering one's love interest so no one else can have them.
The term "yandere" first emerged around 2005 in connection with characters in the adult game "School Days" and the television anime "SHUFFLE!" (which was also originally based on an adult game). There after, there was a surge of similar characters appearing in various manga and anime that came to be known as the "Yandere Boom." This in turn led to much pearl clutching in the Japanese media as older commentators fretted about young fans idolizing these characters as paragons of romantic love when in fact they could also be described as clinically insane.
Since then, the term has become more and more widespread. It has also taken root among overseas anime fans. Most notably, an American game developer called "YandereDev" has begun developing a game called "Yandere Simulator," in which a lovestruck Japanese schoolgirl nicknamed "Yandere-chan" takes it upon herself to eliminate all the other girls connected to her crush. Every time she murders one of them, she becomes more and more insane and disheveled.