[SSJup81]

Kanji: These are pretty much the Chinese characters. A character can represent a word, like the kanji character 光 (Hikari) means light.

Hiragana: This is the Japanese alphabet. Hiragana is pretty much used for words that don't have a kanji character. Hiragana is also only used for Japanese words. Let's say you knew the word for "light" was "hikari", but didn't know the kanji. It would be written in Hiragana as ひかり.

Katakana: This alphabet is pretty much used for sounds and words foreign to Japanese. Like my name is Wendy, but since it's not a native Japanese, it would be written Uendei in Katakana as ウエンデイ as opposed to うえんでい (Hiragana).

Edit: That's the best as I can explain it. Maybe someone else can do a better job.