Discourse refers to connected spoken or written language, i.e. something that is typically made up from more than one sentence (although it doesn't have to be grammatical) and may involve more than one participant. For example:
The type of language used in a discourse can vary depending on context: for example, the vocabulary and grammatical constructs observed in an internet chatroom will be significantly different from those in a government white paper. In some contexts, incomplete sentences may be used, and the sense of a given phrase may depend on previous phrases for its context, e.g. in a conversation where one person answers another's question.
Distinctions between spoken and written forms occur because they cater for different situations: a writer is typically addressing an audience that cannot query what he writes, and in many cases creating something that may be re-read many times. On the other hand a speaker taking part in an informal conversation can interact with his listener and therefore can be less careful about any given utterance being comprehensible, since he can always restate or clarify. Additionally, what he says disappears into the air as soon as it's spoken. Obviously there are different kinds of written/spoken discourse which do not conform to these stereotypes, e.g. an internet chatroom or an audiobook.
Some of the differences between spoken/written discourse from the handout, with explanations, are: