If you're looking to learn the basics of the Japanese spoken language or writing system, this is unfortunately not the page for you. There are many websites out there that can teach you such things. This page is intended to give a brief overview of the evolution of Japanese written language so that someone writing a historical piece would know what sort of writing system his or her characters should use.
Current Japanese Writing Systems
The Japanese writing system currently consists of three character sets. These are the kanji, the set of Chinese characters brought to Japan and used to represent most nouns, proper names, roots of verbs, and roots of adjectives and adverbs. Kanji are not phonetic; you cannot look at a kanji and know what sound you should make to read it, unless you are already familiar with the word it represents. Kanji are symbolic, and often composed of identifiable elements called "radicals", so that if you had never seen a kanji before, you might be able to identify the radicals and make some sort of guess about what the kanji means. In theory, kanji have two pronunciations: the on-yomi or "sound-reading," the Japanese equivalent to the Chinese word for the kanji; and the kun-yomi or "explanatory-reading," the pronunciation of the native Japanese word that matches the meaning of the Chinese word for the kanji. Confused yet? Then you probably don't want to know that most kanji have several on-yomi and several kun-yumi (often two or three of each, sometimes more). Atop that, most kanji have multiple meanings. In my opinion, this is the heart of the great difficulty English speakers have in learning written Japanese.
Most Japanese family names consist of two or three kanji, and most Japanese given names consist of one or two kanji.
The second character set is the hiragana. This is a Japanese-derived phonetic writing system. This makes learning hiragana much easier than kanji, as there are only 46 hiragana, and all of them are pronounced exactly the same way. The individual characters have no meaning, as individual letters in English have no meaning, but when combined, they form words. Hiragana are used for simple words, honorrifics, verb endings, adjective and adverb endings, sentence-helper words called "particles," transitional words, and many other uses.
The third character set is called the katakana. This is also a Japanese-derived phonetic writing system. Katakana are easily identifiable from hiragana with practice; katakana have a lot of sharp angles while hiragana is a relatively soft and flowing style. There are 46 katakana, representing the same sounds as the 46 hiragana. Katakana are used for mimicking the sound of foreign words, and are thus primarily used to write foreign loan words in Japanese. Katakana are also used as a sort of "bold print," and sound effects in manga are almost always written in katakana. Since katakana is considered more "sharp" and easier to read, most Japanese write their names in katakana if they must do so in a small space, where the kanji for their name would be too difficult to read.
The set of hiragana and katakana are collectively known as the kana.
Methods of writing Japanese words in English letters have been developed since the beginning of Western influences in Japan in the 16th Century. The most popular version at this time is the Hepburn system developed in 1867, and certain variants of this system. Other systems are the Japanese-developed Nihon-shiki and Kunrei-shiki. These systems are collectively referred to as romaji in Japan.
Evolution of Japanese Writing Systems
The first attempt at making a Japanese phonetic system came about from a book of Japanese poetry composed about 760 AD called "Man'youshuu," or "Anthology of Myriad Leaves." This book used kanji not to represent the Chinese meanings of those kanji, but to represent phonetic sounds. This led to the man'yougana, a writing system using these characters to represent sounds of the Japanese syllabary.
By around 800 AD, a particular means of representing those man'yougana with short, quick strokes was developed, and used as a pronunciation guide for Buddhist texts written in Chinese. These became the set of katakana. Eventually, these katakana became the official phonetic writing system of Japan. Many official documents were written in kanji and katakana from at least 1600 AD to 1868.
Around the same time, another means of representing those man'yougana, with slow, flowing strokes, was developed, and became known as hiragana. Though I'm not sure if this was its purpose, the hiragana were used as a means for women to write, as it was believed women lacked the intelligence to write katakana or understand kanji.
Such world-famous novels as The Tale of Genji, written by the noblewoman Murasaki Shikibu around 1000-1020 AD, and The Pillow Book by the noblewoman Sei Shonagon around 990-1100 AD, were written in hiragana.
The Japanese military used katakana exclusively early in the 20th Century, and interestingly enough, almost all Japanese military encryption codes used romaji during World War II. (Encrypting the 20 English letters that were used to Romanize Japanese, rather than encrypting the 46 katakana, made for a more robust and easier-to-use code.) So yes, the Japanese found at least some use for romaji.
Except for a few minor changes to reflect shifts in the sound of spoken language, katakana and hiragana have not changed much since 1000 AD.
Other notable phonetic writing systems, which are seldom if ever used today, are hentaigana and kambun.
Kambun is strongly associated with Nara-era writings, and hentaigana is strongly associated with man'yougana. (Some say it's just another name with man'yougana, others say it's a transitional writing form between man'yougana and hiragana.) Hentaigana is sometimes used for storefront signs for certain food shops, to put forth a very traditional feel.