A syllabary is rather like an alphabet: it's a set of written symbols that represent language, but instead of each symbol representing a single sound, it represents a syllable.
A syllable might be just one sound, as in the sound we represent by the letter a, it might be more complicated, as in la or even as in lag.
Syllabaries sometimes have similar symbols for similar sounds, but a pure syllabary will have no link at all between sound and the shape of the symbol, so that the symbol for do will look nothing at all like da or to, for instance.
Japanese uses syllabagrams for some native and many borrowed words. For example mug is written マグ ma gu. The extra sound on the end of the word, the u, actually reduces the number of symbols the syllabary needs: if it wasn't there you'd need different symbols for mug, mub, muk, mud, muk, mun, mum, etc, instead of just one for ma and one for gu.
Other languages, faced with the same problem, sometimes leave out the final sound of a word so that cat, for example, might end up ca.
It's wonderful, isn't it - oh, but I am glad my language has an alphabet.
Word To Use Today: syllabary. Yes, this word comes from the Latin word syllaba, which means syllable. I can't think how you can use it except to say thank heavens English uses an alphabet and not a syllabary.