Trish here, lurking in a different group! (apologies for encroaching, I was just looking at the great examples posted here)
For my money, narrative is defined by Aristotle as having four components, of which the most important is PLOT (the use of linguistic devices to convey causality, surprise, the unexpected, somethign that deviates from the canonical - and most of all peripetea = TROUBLE). There's also chronology (unfolding over time), characters of greater or lesser virtue (heroes, villains etc), a setting (the 'stage' of course). Following this a couple of millenia later, Kenneth Burke proposed the narrative pentad of Act + Actors + Scene + Agency + Purpose. Enough on narrative for now (except that illness is the ultimate 'trouble').
Discourse is the study of language in use. Why this word rather than that word? Why express an observation in numbers rather than words or pictures? Why this particular metaphor? Why formal language not casual language? What is not being talked about in this document and why? Who is setting the rules for what can and can't be said?
Now, for the advanced level task: given the above, what do you think is the intersection between discourse and narrative? And what lies within narrative but beyond discourse and vice versa?
NB see my paper on discourses of telehealth - might be a good one to think about when answering the above question, so we have a specific example.
http://bmjopen.bmj.com/content/2/4/e001574.short