My MA is half linguistics, so for better or for worse, I've got some expertise
here.
A morpheme is, as Penthe pointed out, the smallest meaningful unit. For
example, the word "dog" means 1 dog. "Dogs" means 2 or more dogs.
Therefore, the "s" is what distinguishes the meaning between these two words;
thus, the "s" becomes a morpheme.
In this same example, the "s" can also be a phoneme, because phonemes have to
do with sound, not meaning. "Dog" sounds different than "dogs," so
whether or not the meaning changes, the "s" is a phoneme.
This becomes lots of fun when you have words that are phonologically identical
yet morphologically different, such as the words "deer," or "sheep."
Then, we have my favorite geek word of all time: the MORPHO-PHONEMIC
ALTERATION. :-) An example of this is pluralizing the word "knife,"
which changes to "knives," or "self" to "selves." Rather than simply
adding an "s" to pluralize the noun, the ending phoneme changes to another
phoneme entirely.
Of course, Tolkien violated this linguistic principle when he insisted on the
plural of "elf" as
"elfs."
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"Kiss me, jords!"