OHHHH IT'S ONYOMI VS KUNYOMI TIME!! Okay sorry for the caps but this is a subject I actually think is really cool.
So the official term for this, when a kanji is read one way in some contexts and a different way in others, is usually a case of onyomi vs kunyomi. -yomi, "reading" (from yomu "to read") basically categorizes words into whether you read them with the chinese pronunciation or the japanese pronunciation.
There are a lot of complicated factors that go into this but the easiest way for english speakers to wrap their heads around it is that the Japanese language ended up with multiple sets of vocabulary in the same way that the English language did with the Germanic language/Romance language split; you have one set of vocabulary for common, everyday use that derives from germanic vocab, and then another set of vocabulary for fancy, formal use that derives from Latinate vocab. The classic example is using swine to describe the animal, which the Germanic-speaking Anglo-Saxon farmers and handlers would tend to, and porc to describe the meat, which the French-speaking Normans would eat. But pretty much every word in the English language has some degree of casual-formal duplication of this kind.
Much like European scholars did with Latin, a bunch of vocabulary got imported by the literati of Japan starting around about the 5th century in order to have a different and more formal register to talk about certain topics. This was imported alongside the Chinese writing system at the same time, which then over the next few centuries was shifted into a simpler, non-logographic, syllabic script that became hiragana (the wavy one.)
But that means that a bunch of imported Chinese vocabulary was conventionally written in their original-or-close-to-it hanzi form, and spoken with their original-or-close-to-it Chinese pronunciation. Words of this type have a much higher incidence of long vowels, left over from the Chinese where each character would represent a single complex syllable varied by vowel intonation, compared to native Japanese words which are polysyllabic and hit more consonants.
So, the system has evolved in various directions over time, and now there are a lot of ways that a single character can be read depending on positioning and context, but the broad strokes are: if the kanji is mostly on its own, surrounded by hiragana, it will take the Japanese reading (kunyomi) and if it's part of a compound word, paired with other kanji, it will take the Chinese reading (onyomi.) The more formal the text -- especially notable with medical, scientific, or governmental texts -- the more dense and compacted it will get, with a much higher ratio of kanji compounds to hiragana, and a correspondingly higher dose of onyomi readings.
To leave off with one example: the video above highlights (手), which is 'hand.' When it appears by itself surrounded by simple words like participles, it takes the kunyomi te, like in the sentence 'take my hand' (手を握って, te o nigitte.) But if you are using it as part of a more abstraction concept, like in 手動 -- done by hand, or manual, see how the english term underwent the exact same shift into latinate when it gains a level of abstraction? -- then it becomes part of a compound word with another kanji. Now it takes the onyomi shu, to become part of the compound word shudou. Catch that long vowel! If it weren't part of a compound word, that 動 by itself would take the kunyomi hataraki, for motion or action.
Anyway, onyomi and kunyomi, fun!
