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What is a mora? I tried to read the Wikipedia article that answers this question, but found it difficult to understand.
Ditto with the related LSE question: Is the concept of syllables pronunciation-relevant in languages with mora-based pronunciation?
That's why I decided to ask the question here.
I recently took a phonology course where Mora Theory was briefly mentioned but we didn't go in depth so your question piqued my interest. I did some light googling and I found this set of lecture notes which I think does a pretty good job explaining it:
Ladefoged (1982: 226 in Vance 1987) states that "A mora is a unit of timing. Each mora takes about the same length of time to say" (p.62)
So, if I am understanding the issue correctly, onsets are not assigned a mora because they do not significantly add to the pronunciation time. That is, the time required to pronounce "ah" and "bah" are not significantly different. That amount of time is defined as one mora.
I'm sitting at my computer trying various onset/vowel combinations to see if this is true and anecdotally it seems to check out. (Incidentally, it would be a good idea to make sure no one else is home before trying this yourself)
Now, some things take extra time to pronounce. Long vowels are longer than short vowels. Codas (post-vowel consonants) take more time to pronounce. Thus, such syllables are 2 moras because the syllable can be divided into two basic units, each taking time to pronounce:
I hope this helps clear up your issue.
I'm not sure what aspect of the Wikipedia explanation you find unclear, but here's an analogy:
Imagine the government asks a research group to figure out "how much the Lincoln Tunnel gets used by people to enter Manhattan from New Jersey in a 24-hour period". There are two obvious ways to obtain this measurement; one could count the number of vehicles that enter the tunnel in a 24-hour period, or one could count the number of people riding in the vehicles that enter the tunnel during a 24-hour period. Either metric could be relevant, depending on what the government is trying to focus on. If they care about factors that contribute to street traffic, then the vehicle-counting metric might be more appropriate. If they are more concerned with the actual population density of Manhattan during the day and how much of it is affected by out-of-towners, the person-counting metric might be more appropriate.
In phonology, syllables are like cars and morae are like people. Every car on the road must contain at least one person (the driver), but a car can also contain multiple people. Likewise, every syllable contains minimally one mora, but it may contain two or even three. Certain phonological processes are dependent on syllable count and some are dependent on mora count; some are dependent on both.
Studying Japanese is a great way to become familiar with the concept of a mora. As the section on Japanese in the Wikipedia article mentions, most hiragana symbols correspond to a single mora (an exception is when one symbol is followed by one of the "little" gliding kana ゃ, ゅ and ょ--in this case the two symbols together correspond to a single mora), so that Nippon, which is written with the four hiragana symbols にっぽん, contains four morae. Basically, because the hiragana system gives syllable-final consonants their own symbols (ん for a syllable-final nasal and っ for a geminate consonant that anticipates the initial consonant of a following syllable) but syllable-initial consonants are just encoded as part of a CV syllable represented by a single symbol (な for na, か for ka, etc.), the writing system encodes an asymmetry that is quite common cross-linguistically: nuclear vowels and coda consonants contribute to syllable weight in Japanese but not onset consonants. So na, a, and ka are "light"/monomoraic syllables, and nai, an, kan, and kat (as in the geminated katta) are "heavy"/bimoraic syllables. (At the risk of stretching the analogy too far, onset consonants, when they are present, are kind of like pets that are riding in the car. They don't get counted as people entering the tunnel!)
For an example of a process that is sensitive to both the syllable and the mora in Japanese, consider pitch accent assignment. In the "standard" Japanese dialect, a pitch accent is associated with the initial mora of a syllable. So with three-syllable words, there are three possible accent patterns for accented words--initial, medial, and final accent (of course words can be unaccented as well). It doesn't matter if the syllables are light or heavy. But if the accented syllable is heavy, the high "tone" (trying to stay pre-theoretical here) of the accent is constrained to associate with the initial mora of the accented syllable.
I hope at least part of this explanation clears things up for you a bit! Let me know if there is anything in particular that is still unclear and I will try to improve my answer.
**UPDATE TO ORIGINAL ANSWER:
Just a point of clarification--the system described above for Japanese is a language-specific one. For example, in some languages that count morae, coda consonants do not get counted as being moraic. And, as @jlawler points out in his comment under @acattle's answer, the phonetic realizations of syllable weight contrasts are not the same in all languages (i.e. a heavier syllable--one with more morae--may not take longer to pronounce than a lighter one), so the definition of a mora cannot be dependent on its phonetic behavior. Below is a list of possible moraic segments (in the rime of the syllable) and some example languages that use those criteria for moraicity (taken from Morén's 1999 dissertation, p.16):
Vowels (Khalkha Mongolian, Yidiɲ)
Vowels + Glides (Gumbaynggir)
Vowels + Non-glottal Sonorants (Kwakwala)
Vowels + All Sonorants (Lithuanian, Tiv)
Vowels + All Consonants except Plain Stops (Metropolitan New York English)
Vowels + All Consonants except Aspirated Stops (Icelandic)
All Segments (Latin, Arabic dialects, Aklan, Koya, Imdlawn Tashlhiyt Berber)
Most languages have some kind of word-accentual system, where one position in each word is more prominent than other positions in the same word. The prominent position(s) may have some or all of these properties: higher pitch, longer duration, greater number of contrasts, louder or more forceful pronunciation. When the final property applies, the language is usually said to have stress accent. In these cases the most prominent position is said to have primary stress, and other positions may have secondary stress if they are more prominent than the least prominent syllable, but less prominent than the most prominent syllable.
When studying the word-accent system of a language, you would like to be able to predict which syllable has primary stress, and which if any have secondary stress, and this often involves counting some number of syllables from the edge of a word. In Polish, for example, you count from the end of the word and then stress (in most words) falls on the penultimate syllable. But there are some languages where the counting is complicated by the fact that some syllables (especially syllables with a long vowel or a coda) seem to "count as two" for the purpose of assigning primary and secondary accent. So that your analysis can still be simple and easy to follow, you just invent a new type of timing unit, the mora, and say that a double-counting syllable is made up of two morae, and a single-counting syllable is made up of one mora.
