There’s a lot of confusion among non-Hebrew speakers when first learning the Hebrew alphabet, because most explanations of the letter aleph involve explanations such as “the onset of a vowel at the glottis” (such as the one in Wikipedia)—which may be technically correct, but make it sound more complicated and intimidating to laymen than it really is.
A better explanation is that, in Hebrew and other Semitic scripts, vowels are regarded as infants—they cannot function on their own, and need to be “carried” by a consonant of some sort. So when a vowel starts a word—as in the words Elohim (God), or (light), or indeed, aleph—you need a sign to “carry” it. So that’s what the aleph is—a carrier of a vowel when there’s no other consonant to carry it, or a placeholder for any vowel (for verb conjugation purposes).
When a word ends with a vowel, you need something to indicate that, too—else the reader might think that the word ended with that consonant. In those instances—the letter aleph will serve, if it’s part of the word’s tri-consonantal root—e.g. בורא boré = “creates”, יקרא yiqrá = he will call”. If such a word is followed by another word that starts with a vowel, the result is a glottal stop, to mark the spot where one word ends, and the next begins—e.g. לא אתה (lo atah), or דא עקא (da aqa).
In quasi-phonetic transliterations of Hebrew, the distinction between a vowel that is carried by an aleph and one that is carried by, say, a héh (at the end of a word), or an ayin, a vav or yod, is usually lost. Hence, in a line such as
מצא מצה במצע matza matza bamatza
there is nothing to indicate to the non-Hebrew reader that the three matzas are distinct words in Hebrew, with different roots. Hence the importance of transcription methods such as SimHebrew, which preserve such distinctions.
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(Originally written in reply to a question at Quora.com).
Notes by Autumn Light 