There are four alphabets the Japanese use. Two of them are used in the Inuyasha sample at right. Most of the characters there are hiragana. There are also four kanji characters.
Kanji are borrowed characters from the Chinese character system. The four kanji in this sample are more complicated than the hiragana. Three spell out Inuyasha's name. The other one is verb root of kaeru (the root is "ka-e"). These characters stand for concepts. They can have different pronunciations, depending on what other characters they are near, and they can be combined with each other and hiragana to produce different words.
Hiragana were invented by the Japanese because the Chinese characters weren't enough. Japanese is structured differently from Chinese, so they had to adopt other characters to fully write down their spoken language. Hiragana is what they came up with. Hiragana are much simpler than kanji, and look very smooth and a little curvy. You could, in one sense, call them the Japanese cursive, but only because they look like cursive. Hiragana have no meaning. They, like our letters, only represent a sound, not a concept. The first hiragana character, the cross with the curly-que around it, is the symbol for the sound "a" (as in father). It has no more meaning than that. Hiragana are not exactly like our letters, though. Each hiragana character represents one Japanese syllable, or mora.
You may notice the tiny hiragana looking characters next to the kanji characters. These are call furigana. They are help for young Japanese readers (and inexperienced foreigners!). These characters "sound out" the kanji for the reader, in case they are not advanced enough to know how to pronounce a particular kanji. There are over 10,000 kanji characters. You must know 3,000 to be considered literate in Japanese.