Kanji usually (but don’t always) have two types of readings: one that comes from the native Japanese language, called kun’yomi, and one that was borrowed from Chinese, called on’yomi. Many kanji have multiple kun’yomi and/or on’yomi, and a handful of kanji have only a single reading – either on’yomi or kun’yomi, but not both.

Once you start to get a feel for things, you can generally start to infer which reading goes where. Basically, kanji sitting on their own usually – but don’t always – use kun’yomi, while kanji existing in compound words (called “jukugo”) usually – but don’t always – use on’yomi. Kanji with attached okurigana (hiragana glued on the back to indicate verb/adjective conjugations and whatnot) always use kun’yomi, no exceptions.

Now, for some inexplicable reason, Kouichi teaches the on’yomi as the “reading”, in isolation, and the kun’yomi as part of the vocab. Please don’t ask me why. Your best bet is usually to learn vocab, and the on’yomi will come as a result of that.

So, 刀 is one of many kanji where the kanji on its own forms a meaningful word. Others include 体 (からだ = body), 頭 (あたま = head), 山 (やま = mountain), 川 (かわ = river), et cetera.　So, on its own, 刀 takes its kun’yomi – かたな – and means “sword”. If you stick it in a word with another kanji, say 刀剣, it takes its on’yomi – とう (the whole word is とうけん = swords). Confusingly, it’s also got a handful of nonstandard readings – 竹刀 is read as しない, and 剃刀 is read as かみそり – but I wouldn’t worry too much about those at the moment.