訓読み kunyomi, literally “meaning reading” - These readings come from applying words that existed in Japanese before kanji were imported to kanji. They are the native Japanese words that then got kanji to go along with them.
音読み onyomi, literally “sound reading” - These are borrowed from Chinese. The ancient Chinese had their own pronunciations for the kanji and when the Japanese borrowed the kanji, they also borrowed the pronunciations. But Chinese and Japanese have very different pronunciation schemes, so these don’t sound like ancient Chinese all that much, and Chinese has changed since then so they don’t sound like modern Chinese either.
名乗り nanori - these are basically like kunyomi, but just for names