What is allomorphy?

In linguistics, a morpheme is the minimal unit of meaning in a language. Morphemes include words (roots) and affixes (postfixes, prefixes, infixes, etc.). An allomorph is a variant of a morpheme. 

So when we look at a word like 'cats', we actually have two morphemes used together to signify multiples members of Felis catus (please note that in the examples below I am using phonemic notation for the sounds of the morphemes, but due to the constraints of this format, I am not using IPA for the vowel sounds; instead, I am using English spelling conventions for the vowel sounds even though the forms  enclosed in // should be read as phonemic representations):

{CAT}+{pl} = /kat/ + /-s/ = 'cats'

So, we can see that '-s' is a morpheme that is used to make nouns plural in English. However, it is not the only form of the plural in English. Consider the word 'dogs':

{DOG} + {pl} = /dog/ + /-z/ = 'dogs'

With the words 'dogs', we see an allomorph of the standard English plural morpheme. It can be realized as /-s/ or as /-z/ depending on the phonetic environment of the word to which it is affixed. In an unvoiced environment, it is /-s/ whereas in a voiced environment it is /-z/.

Not all instances of allomorphy are so straightforward, however. To see two more complex situations, we need look no further than some irregular forms of the plural in English.

The plural of 'mouse' is 'mice', not *mouses. This is an umlaut plural in English. It is still the root morpheme plus the plural morpheme, but in this case the plural morpheme is realized as a change form a back vowel to a front vowel in the root:

{MOUSE}+{pl} = /mais/

The change of the back vowel in /mous/ to a front vowel in /mais/ is an allomorph of the English plural morpheme. Another allomorph of the English plural is a null change, as in deer:

{DEER}+{pl} = /deer/

The null change is an allomorph of the English plural morpheme even though there is no actual change to the root word. The reason for this is that morphemes are abstract units of meaning, and they exist even if an allomorph of the morpheme is phonologically null.