A syllabary is a set of written symbols that represent (or approximate) syllables, which make up words. A symbol in a syllabary typically represents an optional consonant sound followed by a vowel sound. In a true syllabary there is no systematic graphic similarity between phonetically related characters (though some do have graphic similarity for the vowels). That is, the characters for "ke", "ka", and "ko" have no similarity to indicate their common "k"-ness. Compare abugida, where each grapheme typically represents a syllable but where characters representing related sounds are similar graphically (typically, a common consonantal base is annotated in a more or less consistent manner to represent the vowel in the syllable).
The Japanese language uses two syllabaries, namely hiragana and katakana. They are mainly used to write grammatical words as well as foreign words, e.g. hotel is ho-te-ru in Japanese. Because Japanese uses a lot of CV (consonant + vowel) type syllables, a syllabary is well suited to write the language. (It is sometimes argued that the Japanese kana should be called moraic writing systems or the like, rather than syllabaries as they are based on morae, not syllables.)
The English language, on the other hand, allows more complex syllable structures, making it cumbersome to write English words with a syllabary. To write English using a syllabary, every possible syllable in English would have to have a separate symbol. Thus, you would need separate symbols for "bag," "beg," "big," "bog," "bug;" "bad," "bed," "bid," "bod," "bud," etc.