The character of a demon or ghost in the form of a girl is a fairly recent, but popular development in the Japanese horror movie scene. The usage can be traced back to a single movie, namely 'Ring' by Hideo Nakata, which came out in Japan in 1998. The 'dead wet girl' became immensely popular after this, and this is why many Japanese horror movies use the trope.
However, where in movies the usage of the ghost/demon girl is quite new, they are in fact a very old part of Japanese culture. Ghosts, deities and demons are a prominent part of Japanese folklore and there are many different kinds and classes. The girls in Japanese horror movies are usually onryō, meaning 'vengeful spirit'. They are a type of ghost - yūrei - believed capable of causing harm in the world of the living, harming or killing enemies, or even causing natural disasters to exact vengeance to redress the wrongs they received while alive.
Though it's not certain what the origin of the onryō is, their existence can be traced back to the 8th century and was based on the idea that powerful and enraged souls of the dead could influence or harm the living people. Often onryō try to kill their enemies and/or use physical torture. However, this is not necessary: in Japanese folklore, some onryō psychologically torment their enemy to avenge themselves. This happens for example in Yotsuda Kaidan, a story about a woman who, as an onryō, targets her husband.
The specific appearance used in Japanese horror movies also stems from folklore. Originally, yūrei didn't have a particular appearance, but classical Japanese dance-drama (Kabuki) resulted in them having a certain set of standard features. They wear a white burial kimono, have wild, unkempt hair and wear face make-up consisting of white foundation with face painting or 'blue shadows'. Especially the kimono and the wild hair are used a lot in horror movies and the make-up is used to varying degrees.
That being said, not every onryō is a woman. It can just as well be a man, and it can be of any age (usually the age the person was when they died). The young boy Toshio in the Japanese horror franchise Ju-On, for example, is also an onryō.
To add on to /u/Falonian 's answer, the girl in the Ring is directly descended from the ghost story of Okiku and the nine plates. The story goes back centuries and has many variations, but a serving girl either breaks an expensive plate or another person frames her by breaking a plate, then she's thrown down a well. In ghost story retellings, Okiku then will slowly emerge from the well, all wet with her dark hair hanging down, counting the plates and when she gets to the ninth, will shriek out on finding it's missing. If you visit Himeji Castle, you can see the well she was supposedly thrown down.
Those dangerous ghosts in Japanese folklore and literature are more often women than not. Zack Davisson's book Yurei: the Japanese Ghost is a good survey of the subject, but if you're not ready to read a whole book, he also has a website with lots of spine-tingling follklore to read through.
Demons in the form of women/ women transformed into demons also go back all the way to the Heian period, and for information on that phenomenon, check out the chapter: Women Spurned: the Revenge of Oni Women in Noriko T. Reider's Japanese demon lore : oni, from ancient times to the present. Reider, a professor and translator of Japanese language and literature published the book under a Creative Commons License, meaning the whole thing is free to read!