Let me have a go answering this...
First of all Hiragana and Katakana are syllibaries, made up of around 50(?) letters each. They represent sounds: a, i, u, e, o, ka, ki, ku, ke, ko... etc.
Kanji, contain an element of meaning not just sound. There are ~2000 kanji designated as "Daily-use kanji" by the Japanese government, though many more are in use.
Now what are they used for?
Hiragana = Most often used for the grammatical elements of the language like showing the conjugation of verbs/adjectives. Also used to write some words, auxiliary verbs, etc.
Example:
着る = kiru = to wear
着ない = kinai =not wear (negative)
着なかった = kinakatta = did not wear (negative past)
In this case the kanji 着 (ki) at the start doesn't change but the hiragana following it does.
Katakana = Most often used for writing loan words of non-chinese origin and foreign names. Also used for emphasis (kinda like italics in English), some onomatopoeia, scientific names, to write some words which have rare kanji or when you've forgotten the kanji, etc
Appropriate example
:
グローバリゼーション = gulōbalizēshon = globalisation, from the English word.
Kanji = Used for the start of most verbs/adjectives, and most nouns including a lot of Chinese loan words. Also used for most personal and place names.
There's other things like sometimes spelling the same word out in hiragana can give a softer tone, as opposed to using the Kanji which would give it a more formal tone.
Think ありがとうございます vs 有難う御座います (= arigatō gozaimasu = thank you)
Hope I haven't missed anything important!