When referring to kanji, 音読み【おんよみ】 and 訓読み【くんよみ】 are two kinds of readings in Japanese, the former being the intended Chinese pronunciations (in Japanese, of course) of the character and the latter being the adapted Japanese pronunciations of the character. Some characters may have very few readings whereas some may have a lot of them.
Generally, 音読み are written in Katakana, and 訓読み are written in Hiragana, the 訓読み usually having some sort of break to separate the kanji from the extra kana involved (called 送りがな【おくりがな】).
Take the kanji 好 (English meaning: fond, liking), for instance. This kanji is read as コウ, この（む） and す（く）.
Another example would be 行 (English meaning: going), which is read as コウ, ギョウ, アン, い（く）, ゆ（く）, and おこな（う）/おこ（なう）.
The 音読み of a kanji is usually seen in kanji compounds (called 熟語【じゅくご】 which I will get to in a later post) while the 訓読み are either seen in verbs or in Japanese readings of a word.
There are literally thousands of kanji in the Japanese language, but fortunately, one needs to know 1,945 kanji to get by in Japan. Still, it is impossible to try and cram ALL of the readings for ALL of the 1,945 common kanji (called 常用漢字【じょうようかんじ】) in one sitting, so it's best if you only learn a few essential readings of a kanji and learn the rest of the readings through context. After all, it'll probably be a while before you run into アン as a reading for 行, right?
ROMAJI version to come if enough people request it.
I should probably update my profile on this site more often...someday