The Japanese use three alphabets. Yup, three. Two are based on syllables, and the third is essentially the Chinese character system.
The Japanese borrowed the Chinese character system many centuries ago. They kept the same or similar meaning for the characters they borrowed, but they changed the pronunciation to whatever the word was in their spoken language. This system is called kanji.
There are somewhere around 10,000 different kanji needed to be considered literate, so this is by far the most learning intensive part of the Japanese written language. I can't do much to introduce it, but I thought I would introduce the kanji number system. Kanji numbers are used as often as Arabic numerals in Japan, even on the covers of manga. It's relatively simple to introduce and actually useful. So, here we go.
The kanji below are the important numerals.
The Japanese number system is a decimal system, like ours, but they also base it on 10,000, where we base ours on 1,000. This can sometimes be a bit confusing.
Unfortunately the Chinese language is very different from Japanese, so many of the aspects of spoken Japanese could not really be represented by kanji. So the Japanese invented a syllable based alphabet from some of the kanji. Hiragana are used to represent the aspects of Japanese that can't be represented by kanji, especially word endings that change with parts of speech. This alphabet is called hiragana. I have a picture containing the hiragana system below. Click on the picture for a larger version.
The hiragana in this chart can be represented by romaji. First take the consonant above the column, then take the vowel to the right. This forms the romaji representation of the character. Therefor the third character down in the second row from the right can be represented by the consonant "k" followed by the vowel "u"...in other words, this hiragana is "ku". The rightmost column is simply vowels alone. The "n*" in the chart is an exception to this rule. Where this character appears there is no vowel, the entire syllable consists of an "ng" sort of sound.
You may have noticed the four hiragana with little symbols. These are not pronounced in English as their corresponding romaji would initially suggest. "tu" sounds like "tsu" in English, "ti" sounds like "chi", "si" sounds like "shi", and "hu" is closer to "fu". They can be represented by either romaji, so be aware that "ti" should sometimes be pronounced "chi".
Adding the little quotation marks to hiragana in the k,s,t and h columns changes the consonant to g,z,d and b respectively. The small circle on an "h" character turns the "h" into a "p".
Putting a small "ya", "yu" or "yo" after hiragana ki,si,ti and ni change the consonants to ky, sy, ty, and ny respectively. "sy" is pronounced "sh" and "ty" is pronounced "ch". Also, if a hiragana with an initial "s" has quotation marks, the "s" becomes "z", and if a "ya", "yu" or "yo" is placed afterward, it becomes "zy", which is pronounced "j". So, ki + little yu = kyu, or si + " + yo = zyo = jo.
Last but not least is little "tu" or "tsu". Placing this before a hiragana doubles the consonant. A double consonant simply has a bit of a pause before it is pronounced. Thus ti + little yo + little tu + to = tyotto = chotto (a little, a bit).
The Japanese also use a separate syllable alphabet for sounding out foreign words, called katakana. It has different symbols, but each new letter corresponds exactly to another letter in the hiragana alphabet. Katakana tend to have straighter lines than hiragana. They are used for sounding out foriegn words and for onomotopoeia. Katakana are handy for the English speaker to learn because most foriegn words are English, so you only have to be able to sound out the word, you don't have to know Japanese to understand the meaning. Click on the picture below for a larger version.
The rules for katakana are the same as those for hiragana, the only differences are that the characters themselves are different, and katakana has a few extra symbols to try to deal with the many sounds that exist in foreign languages, but are not represented by traditional katakana.
A common example of the additions that are being made to Japanese is representing the English sounds "tea" and "dee". This is done by writing "te" (or "de") followed by a small "i". Other similar small changes can represent other sounds with modified katakana. If you see one of these newer representations, it should be pretty easy to figure out.