Morphology is a study of words. It basically deals with word formation, examines the relationship between words, and analyzes their constituent elements.
Morpheme is the smallest unit of a word, which has a meaning, lexical or grammatical, and cannot be divided into smaller units. For instance, the word "unpresentable" consists of 3 morphemes -- un + present + able. Un is a prefix, which means "not" and is used in this example to negate the adjective "presentable." The suffix able is used to form adjectives and is usually placed at the end of a verb (useable, loveable, deniable, etc.).
Morphs form morphemes, and they are "an element of speech or writing that represents and expresses one or more morphemes" (Microsoft Encarta Dictionary). Morphs are the real forms utilized to form a morpheme (Yule 71 ). For instance, "students" consists of 2 morphemes, student + suffix s. Students is comprised of "one or more morphs" that constitute the "environment" (SIL) of the word students. Students has 2 morphemes, each with 1 or more morphs. The morpheme -s has 1 morph and can have 1 of 2 allomorphs of pronounceable realization: s or z, as in cats (s) and shoes (z).
Allomorphs are different realizations of one morpheme (Yule 72). For example, the words cats, dogs and buses all contain a plural morpheme, and we can deduce that the pluralizing morphemes (-s and -es) have 3 different potential pronunciations - /s/, /z/ and /iz/. We call these allomorphs because they represent different pronunciation potentials of the same morphemes, the plural morphemes (and one plural morpheme has one morph -s, while the other has two morphs -es).
Yule, George. The Study of Language. Cambridge: Cambridge UP, 1996. Web.