Discourse is one of the four systems of language, the others being vocabulary, grammar and phonology. Discourse has various definitions but one way of thinking about it is as any piece of extended language, written or spoken, that has unity and meaning and purpose. One possible way of understanding 'extended' is as language that is more than one sentence.
Something as short as two phrases in a conversation or as long as an entire extended essay are both examples of discourse and both show various features of discourse.
In the classroom
Areas of written and spoken discourse looked at in language classrooms include various features of cohesion and coherence, discourse markers, paralinguistic features (body language), conventions and ways of taking turns.