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.
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.
Other languages that use syllabic writing include Mycenaean Greek (Linear B) and Native American languages such as Cherokee and Cree. Several languages of the Ancient Near East used Cuneiform, which is a syllabary with some non-syllabic elements.
The Indian languages and the Ethiopian languages have alphabets (called abugidas by some scholars) that look like syllabaries to western eyes, but are not. They both use separate consonant and vowel signs. Most often, the vowel sign is added to the consonant sign which may give the impression of a syllabic unit.
See also Writing Systems.