The Kana (Hiragana and Katakana), and Kanji

There are two sets of kana, each set containing 46 letters. One set is called Hiragana and the other called Katakana, and each set contains identical sounds to the other. The Hiragana are mostly used for grammar and words with no kanji (or where the author doesn't know the kanji), and also for slang, and sometimes for very common words, though this varies from author to author.. The Katakana are mostly used to write foreign words in Japanese, and for emphasis - similar to using italics.
Each kana is a syllable rather than a letter and most kana are combinations of one consonant and one vowel. The syllables are formed basically by adding each of the vowels (A, I, U, E and O) to each of the consonants (K, S, T, N, H, M, Y, R and W). The vowels and N each have their own sounds and kana by themselves. There are a few exceptions though.
The sounds and kana can be modified by adding two small lines (called nigori) or a small circle (call maru) to certain of the kana. For nigori, K to G, S to Z, T to D and H to B. For the maru, H to P.
The pronunciation of the vowel parts can also be lengthened by adding certain of the vowel kana after another kana. For example:

TO + U becomes TÖ, which is pronounced just like 'to' but with a longer 'o'.
Also, some kana are written combined in a special way that modifies the pronunciation - by adding a ya, yi or yo after a '[consonant]i' it becomes '[consonant];y[a,i or o]' (with one or two exceptions - see the charts). An example:

KI + YO becomes kyo. The ya, yi or yo part is written with a smaller character to differentiate it between when you want KIYO.

Also, the small version of the tsu character place doubles the pronunciation of the next character. For Example:

(tsu) + TA becomes TTA. It also sometimes used to indicate an abrupt finishing of the sound at the end of a sentence, or subsentence. For example:

Gives Aa! Megumi-sama.

Kanji
The Kanji are the Chinese ideographic characters the Japanese borrowed so that they could write about 2000 years ago. Today, the same character (or similar, as there has been some divergence in the drawing style) has similar meanings, though different pronunciations. However, many Kanji have an on pronunciation which is based on the original Chinese pronunciation, and they also have one or more kun pronuciations (orignal Japanese) for when the kanji is used in compounds. Most kanji have more than one pronunciation - at least on when used as a word by itself, and another when used in compounds, though some characters have many pronunciations.