Morpheme is the mental realization of various allamorophs defining it and morphs are concrete realization of these allamorphs . For instance, the plural /s/ morpheme has three allamorphs represented the morphs /s/ , /z/, and /iz/ as in books, bags, and dishes respectively.

Morphemes are the smallest meaningful constituents of words; Words are composed of morphemes (one or more). sing-er-s, home-work, un-kind-ly, flipp-ed, de-nation-al-iz-ation nej-ne-ob-hospod.ar-ova-tel-nejˇs-´ıho, auto-servis-u Morph. The term morpheme is used both to refer to an abstract entity and its concrete realization(s) in speech or writing. When it is needed to maintain the signified and signifier distinction, the term morph is used to refer to the concrete entity, while the term morpheme is reserved for the abstract entity only.

All contributors are right, in a way. Still, we must bear in mind that linguistic categories are - unavoidably - theory-dependent. Edward Sapir (whom I still admire) used a completely different model of grammar, compared with, say, Distributed Morphology. Therefore, the answer to the question will show differing directions of bringing morphemes and morphs together. I've always had a problem with notions such as "abstract entity": assuming a "morpheme"{NOMINAL.PLURAL} for English, this is a pushover for the agglutinative /s/-Plural. But what about the "irregular" plurals: children, mice, sheep etc.? Do they show allomorphs of the same {NOMINAL.PLURAL} morpheme? Or, are there additional {NOMINAL.PLURAL} morphemes? If so, how many? This amounts to the question of where and how morphemes do exist. If (as I assume) they are a mere linguistic (= linguists') construct, the claim of their existence is of little avail. Entia non sunt multiplicanda praeter necessitatem (even if William of Ockham never used this wording).