What Are Morphemes in English?

Definition:

A meaningful linguistic unit consisting of a word (such as dog) or a word element (such as the -s at the end of dogs) that can't be divided into smaller meaningful parts. Adjective: morphemic.

Morphemes are commonly classified into free morphemes (which can occur as separate words) and bound morphemes (which can't stand alone as words).
See Examples and Observations, below. Also see:

Affix

Etymology:
From the French, by analogy with phoneme, from the Greek, "shape, form"

Examples and Observations:

A prefix may be a morpheme:
"What does it mean to pre-board? Do you get on before you get on?"
(George Carlin)
Individual words may be morphemes:
"They want to put you in a box, but nobody's in a box. You're not in a box."
(John Turturro)
Contracted word forms may be morphemes:
"They want to put you in a box, but nobody's in a box. You're not in a box."
(John Turturro)
"A word can be analyzed as consisting of one morpheme (sad) or two or more morphemes (unluckily; compare luck, lucky, unlucky), each morpheme usually expressing a distinct meaning. When a morpheme is represented by a segment, that segment is a morph. If a morpheme can be represented by more than one morph, the morphs are allomorphs of the same morpheme: the prefixes in- (insane), il- (illegible), im- (impossible), ir- (irregular) are allomorphs of the same negative morpheme."
(Sidney Greenbaum, The Oxford English Grammar. Oxford Univ. Press, 1996)

Grammatical Tags
"In addition to serving as resources in the creation of vocabulary, morphemes supply grammatical tags to words, helping us to identify on the basis of form the parts of speech of words in sentences we hear or read. For example, in the sentence Morphemes supply grammatical tags to words, the plural morpheme ending {-s} helps identify morphemes, tags, and words as nouns; the {-ical} ending underscores the adjectival relationship between grammatical and the following noun, tags, which it modifies."
(Thomas P. Klammer et al. Analyzing English Grammar. Pearson, 2007)
Language Acquisition
"English-speaking children usually begin to produce two-morpheme words in their third year and during that year the growth in their use of affixes is rapid and extremely impressive. This is the time, as Roger Brown showed, when children begin to use suffixes for possessive words ('Adam's ball'), for the plural ('dogs'), for present progressive verbs ('I walking'), for third-person singular present tense verbs ('he walks'), and for past tense verbs, although not always with complete corectness ('I brunged it here') (Brown 1973). Notice that these new morphemes are all of them inflections. Children tend to learn derivational morphemes a little later and to continue to learn about them right through childhood . . .."
(Peter Bryant and Terezinha Nunes, "Morphemes and Literacy: A Starting Point." Improving Literacy by Teaching Morphemes, ed. by T. Nunes and P. Bryant. Routledge, 2006)