Morphemes are form/meaning pairings (where "form" = distinctive string of sounds, and "meaning" includes both meaning in the usual sense, and function). Morphemes can be roots or affixes, depending on whether they are the main part or dependent part of a word (cf. Roots vs. Affixes).
It is important to recognize that there is no one-to-one correspondence between form and meaning, and that what counts for identification as a morpheme is both form AND meaning. Let's consider some potentially tricky situations that can arise in deciding whether we're dealing with a single morpheme or more than one:
1. Two different morphemes can accidentally have the same form. Some English morphemes for which this is the case are the following ("Greek prefix", "Latin root" etc. are abbreviations for "prefix borrowed from (Classical) Greek", "root morpheme borrowed from Latin" etc.):
The unrelatedness of the meanings tells us they are different linguistic units. There is no psychological connection between them, and typically their origins are completely different.
These characteristics (same sounds in the units but no relation between the units) make these cases examples of homonyms. Homonyms are familiar to most English speakers in examples like bank 'financial institution' and bank'riverside'. These cases, though, happen to be bound rather than free morphemes.
2. Forms with the same meaning may also be different morphemes. There are two subcases of this:
(a) the forms may be rather different from one another. Example:
In this example, the first two morphemes were borrowed into English from different languages, a sufficient reason for thinking of them as different elements and hence distinct morphemes. The third is native English, which means another different linguistic source and hence a different element. It so happens that in this case, all three morphemes go back to a prehistoric word meaning 'not' that linguists have reconstructed as part of the original language that gave rise to Latin, Greek, English, and other related languages. But the connection is too far back to think of them as a unitary element in English.
(b) the forms may be the same or very similar, but like the above case, their sources are different languages. Example:
As above, these two happen to go back to a common ancestral source morpheme, before Latin and English (and their closest relatives) evolved into separate languages. (This historical fact accounts for why the forms are similar.) But again, the unity of these elements is only historical. Because the immediate source languages are different, it is reasonable to think of them as different elements. This kind of situation, in which our definition of morpheme as an element pairing a particular form with a particular meaning might lead us to call these one morpheme, but our historical knowledge leads us to call them two, is comparatively rare. We need not let such a borderline case detract from our basic understanding of a morpheme. They are mentioned here only for completeness' sake. A somewhat similar case is the prefix en- as in enlarge and engross, which is from Old French. Historically en- is from Latin (since Old French developed out of Latin); the vowel of the original in- was lowered in the mouth to yield en-, but the meaning of 'in, into, intensifier' remained similar. We thing of them as distinct morphemes because their immediate source is different languages, as in the case of Latin and Greek, even though the elements are cognates.
3. Two forms with the same meaning may be alternate forms of the same morpheme. Example:
In these cases, the two forms are very similar, often differing in one consonant or vowel. They typically result from a situation in which an original single form adapted its beginning or ending sounds to the sounds found in other morphemes it combined with. Often there is some pattern to the alternation between the two forms (e.g. the Greek 'not' morpheme is found in the form a- before roots beginning with consonants, and an- before roots beginning with vowels.)
The alternate forms in these cases are called allomorphs ( < Greek prefix allo- 'other'). We will discuss many cases of allomorphy in class; they are treated in Chapters 4 and 6 of our textbook.