One way to treat differences in inflectional morphemes is by proposing variation in morphological realization rules. In order to do this, we draw an analogy with some processes like in phonology.
If we consider ‘phones’ as the actual phonetic realization of ‘phonemes’, then we can propose morphs as the actual forms used to realize morphemes. Thus, the form cat is a single morph realizing a lexical morpheme. The form ‘cats’ consists of two morphs, realizing a lexical morpheme and an inflectional morpheme (‘plural’). Just as we know that there were ‘allophones’ of a particular phoneme, then we can recognize allomorphs of a particular morpheme.
Take the morpheme ‘plural’. Note that it can be attached to a number of lexical morphemes to produce structure like ‘cat + plural’, ‘sheep + plural’, and ‘man+ plural’. Now the actual forms of the morphs which result from the single morpheme ‘plural’ turn out to be different. Yet they are all allomorphs of the one morpheme. It has been suggested, for example, that one allomorph of ‘plural’ is a zero-morph, and the plural form sheep is actually ‘sheep + o ’.