Morpheme

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In Morpheme-based morphology, a morpheme is the smallest language unit that carries a semantic interpretation. Morphemes are, generally, a distinctive collocation of phonemes (as the free form pin or the bound form -s of pins) having no smaller meaningful members.

English example: The word "unbelievable" has three morphemes "un-", (negatory) a bound morpheme, "-believe-" a free morpheme, and "-able". "un-" is also a prefix, "-able" is a suffix. Both are affixes.

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Types of morphemes

Free morphemes like town, dog can appear with other lexemes (as in town-hall or dog-house) or they can stand alone, or "free". Allomorphs are variants of a morpheme, e.g. the plural marker in English is sometimes realized as /-z/, /-s/ or /-Template:IPA/.
Bound morphemes like "un-" appear only together with other morphemes to form a lexeme. Bound morphemes in general tend to be prefixes and suffixes. Morphemes existing in only one bound form are known as "cranberry" morphemes, from the "cran" in that very word.
Inflectional morphemes modify a word's tense, number, aspect, and so on. (as in the dog morpheme if written with the plural marker morpheme s becomes dogs).
Derivational morphemes can be added to a word to create (derive) another word: the addition of "-ness" to "happy", for example, to give "happiness".