Syllables vs Morae
In English, we know the syllable as the most common unit that we break speech sounds into: a vowel sound, sometimes preceded, followed by, or surrounded by consonants. We're taught that there are only five vowels in English -- a, e, i, o, and u -- but in reality, depending which form of English is being spoken, there can by anywhere from fourteen to twenty -- arguably more. Similarly, while there are 21 non-vowel letters that represent consonants in the alphabet, there are actually 24 consonants in the language.
Examples of English syllables:
...and so on.
There are syllables in Japanese words, as well:
However, syllables are not what Japanese words are commonly broken down into, but rather, morae (singular mora). These are known by Japanese poets as on 音, phonetic units (Japanese linguists, however, would use the term haku 拍 or the transliteration of the word mora itself, mōra モーラ). A mora can be considered something equivalent to a micro-syllable, of sorts. The Japanese syllabary itself is based on morae, despite the name -- each kana symbol equals one mora. Morae are very rarely heard in Japanese speech, but they are discernible in many, many songs both modern and ancient, and almost all classical Japanese poetry relies upon mora-count, not syllable count.
For an example of mora in action, take chō 蝶 and kintsugi 金継ぎ from the above list. While these words have one and three syllables, respectively, they have two and four mora to them:
And even better example of this in action is the fantastic word onmyōdō 陰陽道 -- 'the path of yin-yang':
Despite the fact that morae do exist within English, they are both ridiculously complicated to explain and define -- even for linguistic scholars! -- and kind of...well, frankly, uninteresting and mostly unimportant. In English, the syllable really is the king of sounds, which is why to compose Japanese poetry forms in English, the poetry community and linguists alike largely find it a more than adequate equivalent to mora -- it is literally as close as we can get to it.

A Quick Note...
I do not use very technical linguistic language in this essay, being that I want everyone to be able to grasp this concept without needing to have much linguistic knowledge at all. If you are a linguist/student of linguistics, you may think I'm using oversimplified terms, and speaking as a linguistics student myself, I certainly am! This essay isn't meant to be complicated; the layman should be able to read through this and come away with a decent understanding of syllables and morae, nothing else. The extras aren't needed when you're simply laying a foundation. ♥