Morphs and allomorphs are both ways to describe the phonetic expressions (that is, the actual sounds produced) of the smallest meaningful parts of language in the study of morphology: morphemes.
A word like "unhappy" has two meaningful pieces: "un" and "happy," which together convey the meaning "not happy." Each of the meaningful parts is a morpheme, the phonetic expression of which is called a morph. Morphs can be further classified into "lexical" or "grammatical": lexical morphs are the meaningful roots, like "happy" or "man," and these are often free-standing words. Grammatical morphs modify the root in a meaningful way, but may not stand as a free word; in English, suffixes like [-able] or prefixes like [un-] are grammatical morphs.
Allomorphs are phonetic variants of a morph. A good example of allomorphy is the plural suffix in English, which can have the allomorphs [-s], [-z], or [-ez] depending on the phonetic environment.
One way of looking at this is that any allomorph is simply what you call a morph that has another possible phonetic expression.