morpheme
/ MOR-feem /
A morpheme is the smallest piece of language that still carries meaning — the atom of word-building. Take the word cats: it splits into cat (a furry animal) and -s (more than one). You can't chop cat any smaller and still have anything that means something — cut it into c-a-t and you're left with bare sounds, no meaning at all.
Words are built by snapping these pieces together. A root like teach can pick up -er to become teacher (one who teaches), then -s to become teachers. Unhappiness stacks three at once: un- (not) + happy + -ness (the state of). Some morphemes stand alone as whole words (cat, run, blue); others, like -s, un-, or -ness, can only cling to something else.
Here's the catch people miss: a morpheme is not the same as a syllable. Crocodile has three syllables but is just one morpheme — none of its chunks means anything on its own. And cats, with one syllable, holds two morphemes. Meaning, not sound, is what marks the boundary.
un- + happy + -ness → unhappiness (3 morphemes, 1 word)
A root plus two affixes — meaning stacked piece by piece.
The term was coined in the late 1800s from the Greek morphē, "form," on the model of phoneme. Morphemes that stand alone are called free (cat, blue); those that must attach to a host are bound (-s, un-, -ness).