OED gives as meaning b.: "phonological unit of language that cannot be analysed into smaller linear units and that in any particular language is realized in non-contrastive variants. Also attrib ... Although its exact nature is disputed, and the existence of an abstract phonemic level (and hence the abstract phoneme as a constituent of morphemes) is controversial in phonological theory, the phoneme remains a standard taxonomic unit in the description of speech. The phoneme of structural linguistics is sometimes called the autonomous or taxonomic phoneme by generative phonologists, and distinguished from the systematic phoneme."
This should be enough to warn you that the word phoneme is disputed by experts. However the word is very useful to those who are teaching, and the concept to those who are learning, languages. You do not need to understand the detail of OED's note above.
To simplify the definition: a phoneme is a unit of sound in language. It is a label for the smallest unit that can make a difference between two words in that language. (A phoneme is one of a minimally contrastive pair.) In English, for example, the words '˜bat', 'bet', 'bit' and 'but' are very clearly different words. But they share the same first and last sounds. The difference between them lies in their vowels. So we can say that the sounds '-a-', '-e-', '-i-' and '-u-' are distinct phonemes. If we consider 'bat', 'cat' and 'fat' we can see similarly that the sounds '-b-', '-c-' and '-f-' are phonemes, and also that '-n-' and '-r-' can be added to the list ('ban' and 'bar', as well as 'can' and 'car', 'fan' and 'far').
For more information about phonemes, see phoneme - more information.