Since it's made up from a variety of dialects about to disappear, japanese hasn't been classified; and its origin is rather unkown.
It's often related with korean, although both languages have nothing to do with each other.

   In the early third century, the close cultural relationship between Japan and China lead to the inclusion of the chinese written symbols. But their phonetic value was given more relevant than their actual meaning.

   This style of writing was called “maniogana” from which the modern style of writing emerged. Later, in the IX century, those chinese symbols (kanji) were simplified until they formed a system based on 48 symbols known as “kana”.

   Kana was divided in two: Katakana and Hiragana. Hiragana is more common than katakana, mostly because hiragana was used by noble ladys during the Heian period (VIII-XII centuries) and hiragana was chosen also to write many poems and tales.
As a side note, men preferred kanji.

   Since the XV century, katakana has been used to write practical and technical text, and hiragana is preferred in literature. After the Meiji restauration, the dialect of Tokyo became the base of the common written language.

   The actual writing system still uses kanji and hiragana; but its spelling has changed dramatically after WWII. Modern kanji now has 1945 symbols and Katakana is used just for typographical matters. Modern japanese is written in vertical columns right-to-left, but a more “western” left-to-right flow is used sometimes.