Kanji are Chinese characters used to write Japanese words, or parts of words. Unlike kana, Kanji represent meanings, and how you pronounce them depends on the word they are used in. There are between 2000-3000 kanji symbols that are commonly used in Japanese writing.
The Jouyou Kanji is a list of Kanji containing 1945 characters which Japanese students learn up until they graduate high school. In Japan, it is expected that you know the Kanji in this list after high school as they are necessary for daily use.
Jinmeiyou Kanji is a list of Kanji that are used for Japanese names. There are 985 characters in this list.
All characters outside the Jinmeiyou and Jouyou list are called Hyougaiji. The exact amount of Kanji in this list is unknown. These are kanji characters that are used for non-common words or words where hiragana or a different kanji is now prefered. Often you will see hyougaiji used in special art and poems.
Furigana are small kana printed above Kanji in order to help the reader know how to pronounce the kanji. They are often used in children's media, manga, karaoke lyrics, uncommon kanji, maps and railway stations. Furigana is also used for the hyougaiji since their meanings are not widely known.