nice gaijin said:
Katakana is used for words that are not native to Japanese (with few exceptions). There are some cases where much more is written in katakana, such as manga.
Katakana can also be used to put emphasis on any word and it's often used this way in advertising and also manga, as nice gaijin mentioned.
As for furigana, furigana are small hiragana and/or katakana that are placed by kanji to give the pronunciation (they are written above the kanji when writing horizontally, and on the right of the kanji when writing vertically). It is often used in children's books and manga aimed at kids and teens, who are still learning kanji in school. Sometimes furigana is used to give a double meaning too, especially in song lyrics where, say, the word 女友達 (onna tomodachi, female friend(s)) is written in one line but the singer sings 友達 (tomodachi, friend(s)) instead, and likewise in the written lyrics ともだち is written above 女友達.
And yes, manyougana is long obsolete - it was a system of using only kanji to write Japanese, and it was used before hiragana and katakana were invented.
Hentaigana were alternate ways of writing certain kana (the "hentai" in hentaigana is not the "hentai" you probably know about
The "hentai" in "hentaigana" in this case means something like "variant" or "abnormal" and it's written 変体), and this is generally obsolete as well, though some soba restaurants still use them on their signs to give an archaic feeling, like writing "Ye Olde Ale Shoppe" or whatever in English. Most Japanese people can't read most hentaigana nowadays though.