In linguistics, Marina’s definition is right on – a “discourse” is a lengthy unit of language, longer than a sentence.
In philosophy, “discourse” also means a conversation or discussion or debate involving specialized knowledge. It can also mean an institutionalized approach to such a debate—especially if the approach places limits on what you can say and remain socially acceptable.
As an example, in my own academic field, musicology, there are two approaches to musical analysis. One of them is purely structuralist—we accept a priori that Beethoven’s 9th Symphony is great art, and then look at how all the pieces fit together. We don’t look at references to anything outside the notes themselves. The other is post-structuralist—we consider what Beethoven’s 9th Symphony means to the people who heard it, and how extramusical references convey the meaning of the symphony.
These are two different discourses, because the people in the structuralist camp think that any reference to, say, masculine motifs and feminine motifs is invoking irrelevant things from outside the music—despite that 19th century music analysts and commentators definitely heard “masculine” and “feminine” elements. And the people in the post-structuralist camp get irritated at the limitations of the structuralist approach—we don’t love Beethoven’s 9th because of all the structural things in it, but because it speaks to us, and any analysis that ignores that is about as interesting as algebra homework exercises.
And narratives are stories. When we take a series of events, and filter and reinterpret them so that their ordering has meaning, we make a narrative out of them. Doing this is one of the ways that people make sense out of events in their lives.