A useful definition of morpheme–good enough for most purposes–is “a minimal and indivisible morphological unit that cannot be analyzed into smaller units.”
This broad definition is adequate for most general discussions, but it’s possible to get more specific. Just for fun, here are four different kinds of morpheme.
allomorph or morph: any part of a word we want to talk about. A morph can be a whole word, like dog, a meaningful affix, like un- or -ness, or a part that has no meaning, but is separable, like the o in kissogram (a telegram delivered with a kiss, intended to amuse or embarrass the recipient.)
portmanteau morph: a single form which consists of two or more morphemes, but which cannot be divided neatly. For example, the verb crashed can be separated into the morphemes crash and -ed, but a word like sang, which consists of the stem sing and a past tense marker (the changed vowel), cannot be so divided.
empty morph: a piece of a word that does not contribute to its meaning, but is necessary to make it easily pronounceable. For example, the o in kissogram. (Linguists argue about something called a “null morpheme,” but as I’m not writing for linguists, I won’t go there.)
cranberry morpheme: a morpheme that occurs in only one word, like the cran in cranberry, the twi in twilight, and the -art in braggart.
Note on cran-, twi-, and -art:
The Cape Cod Cranberry Growers’ Association explains the cran in cranberry this way:
The name “cranberry” derives from the Pilgrim name for the fruit, “craneberry”, so called because the small, pink blossoms that appear in the spring resemble the head and bill of a Sandhill crane.
The OED entry tells us that the word cranberry was unknown to herbalists writing in the 16th and 17th centuries. They knew about cranberries, but they called them other names, such as marsh-whorts and fen-berries. The North American cranberry growers may have adopted the name from German immigrants. Low German has the forms krônbere, krones- or kronsbere, krônsbär, kranebere; all translate as “crane-berry.” British speakers adopted the word cranberry in the 18th century.
This allomorph may also derive from German. Both High German and Low German have words that mean what twilight does. Middle High German has zwischenliecht, “tweenlight”; Low German has twêdustern, twêdunkern, literally “twi-dark.”
This affix belongs to class of suffixes that turn a verb into a noun doer of the action. The spelling -ard at the end of words entered English from old French in such words as bastard, coward, mallard, and wizard. The -ard, sometimes spelled -art, became a formative of English derivatives with the sense of “one who does to excess, or one who does what is discreditable.” Examples of words in which –ard conveys the discreditable connotation include drunkard, laggard, and sluggard. Braggart is the only -art survivor in common use.