The Japanese writing
system is a remarkable example of how one culture
has assimilated part of another, and adapted things to its needs.
Originally, when the Japanese first came into contact with the Chinese civilisation, they had no writing system; which goes to show that writing is not essential for civilisation. The educated and the well-born would learn Chinese as the lingua franca, the Latin of the eastern world.
First recorded Japanese contact occured during the Three Kingdoms period of Chinese history, around 220 - 265 CE. Queen Himiko sent an envoy to the Kingdom of Wei, with its legendary capital at Luoyang. After the collapse of the detente of the Three Kingdoms (Wei, Wu, and Shu), China underwent a period of instability, and Japan took the opportunity to establish the colony of Mimana on the Korean peninsula. Mimana was the principle point of contact between the two cultures until its loss around 560 CE.
Chinese is an language which writes with ideographic characters. Each character represents one idea, and one sound. The Japanese language shares no common features, and is polysyllabic. At this time, the kana syllabaries, hiragana and katakana, did not yet exist.
Buddhism was also instrumental in the transmission of language to the Japanese. When Buddhism entered China, the names of the Deities in sanskrit were transliterated into Chinese sounds. From there, into Japanese, Chinese sounds were preserved, and came to be known as Go-on (Sounds of Wu, Wu being one of the Three Kingdoms, and Go being the Japanese reading of the character).
At this time, Chinese characters were imported along with their meanings in their entirety, and Japanese readings were attached to words. Due to the intrinsic differences of languages, Japanese scholars came onto the idea of transliterating Japanese sounds using Chinese characters. This system is known as kana, meaning "false name". Some of the earliest known classical works use this system, including the Manyoshu (the thousand leaf collection), from which the system also derives the name "Manyogana", or "kana of the Manyo"
Obviously, there was great difficulty in being able to tell what word was part of a compound, and consisted of which sound, as all writing was still in kanji. Katakana developed as a way of indicating pronunciation, and was placed beside the kanji to help the reader.
Scholarly elites and educated monks studied classical Chinese texts, often in the original. The literature developed in Chinese and in Manyogana was called kanbun, or Chinese literature.
Hiragana was developed mainly by women, and they also wrote many notable works. Because of this reason, it was originally known as onnade, or woman's hand. Literature liberated from kanji by hiragana was known as wabun, or Japanese literature.
The writing of hiragana derives from the "grass" style of Chinese calligraphy, and from the characters used to fix sounds in the Manyogana.
In Japanese, the Chinese characters often have many multiple onyomi, resulting from the importation of characters of the period of many Chinese dynasties, where spoken language was in a constant flux. Although the standardised system of writing in China dates from the first unification of China as a country, spoken words often change sounds between dialects, and between the standard Chinese of the day. Apart from the original Go-on of the Three Kingdoms period, Tou-on comes from the Tang Dynasty; this dynasty was during a great flowering of Chinese arts and intellectual acheivements. Tou-on sounds persist in many Chinese words in dialects today. Also, Sou-on is the classification of sounds imported since the Sung dynasty. Another class of readings, Kanyou-On derives from Japanese mispronounciations. The word for bronze originally had the reading tou, which was corrupted to dou.
As an additional correction to Jeeve's writeup, as a consequence of Japanese's long and venerable history of association with Chinese characters, almost all words can be written using kanji of some sort, including but not limited to "aru", "suru", "oru" etc (as a consequence, also "desu", "da", etc.). Although not common, it may be seen (and used).