The continuous morpheme is an uninterrupted string of phonemes building up a morpheme.
The discontinuous morpheme is a grammatical unit built up of an interrupted string of phonemes. It is seen in the analytical grammatical form comprising an auxiliary word and a grammatical suffix.
Example:
be ... ing — for the continuous verb forms (e.g. is going);
have ... en — for the perfect verb forms (e.g. has gone);
be ... en — for the passive verb forms (e.g. is taken)
To sum up:
The morpheme is a recurrent meaningful from which cannot be further analyzed into smaller recurrent meaningful forms. It is syntactically or positionally bound and cannot take any arbitrary position. It never expresses both a lexical and a grammatical meaning. Lexical morphemes have consistent meaning beyond whatever grammatical information they also carry. Grammatical morphemes, on the other hand, function only to express grammatical meaning. Grammatical meaning is recurrent and systemic, it is general and abstract, while lexical meaning is free and independent, concrete and material.