The お名前 (onamae) field is for your name as written in Japanese normally, and it should take any of kanji, hiragana, or katakana as appropriate so long as it is 全角 (full width) and the programmer didn't screw up.
The フリガナ (furigana) field is to indicate how the name given above is read, because kanji can have many readings including completely arbitrary ones. This also serves to indicate how computers should index and sort the names when storing and processing them, so it's still applicable even if the name is all hiragana or katakana and immediately obvious.
Furigana is also used to indicate how to read kanji in ordinary text, oftentimes when dealing with rare kanji or special readings, when the text must be comprehensible by everyone (eg: emergency bulletins), or when the text is written for people learning Japanese (eg: school textbooks).
Furigana is usually written using hiragana, so the reason 全角カタカナ (full width katakana) is specified instead of just full width is to inform the form's filer that he shouldn't write hiragana like he otherwise probably would.