In Western classical rhetoric

Rhetoric

Rhetoric is the art of discourse, an art that aims to improve the facility of speakers or writers who attempt to inform, persuade, or motivate particular audiences in specific situations. As a subject of formal study and a productive civic practice, rhetoric has played a central role in the Western...

, the exordium was the introductory portion of an oration. The term is Latin

Latin

Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

 and the Greek equivalent was called the Proem or Prooimion.

The exordium is one of six parts of a discourse that an orator would develop as part of the rhetorical discipline known as dispositio

Dispositio

See also: Disposition Dispositio is the system used for the organization of arguments in Western classical rhetoric. The word is Latin, and can be translated as "organization" or "arrangement."...

 — the arrangement of the arguments in an oration. In the exordium, the orator laid out the purpose of the discourse. In doing this, he would need to consider several things:

• What kind of cause is he presenting? For instance, is it an honorable cause (defense of a hero) or a dishonorable one (defense of a murderer)?
• Should a direct opening be favoured, or should the opening be more subtle and indirect?
• In what manner ought the speaker to proceed (e.g., light-heartedly or seriously)?
• The speaker should introduce their own character or credentials, so as to make the audience predisposed to believing their arguments.
• If required, or possible, the speaker might also call into question the character or credentials of his opponent.
• Lastly, the speaker must avoid certain faults in the introduction. For example, this excerpt from the Rhetorica ad Herennium
  Rhetorica ad Herennium
  The Rhetorica ad Herennium, formerly attributed to Cicero but of unknown authorship, is the oldest surviving Latin book on rhetoric, dating from the 90s BC, and is still used today as a textbook on the structure and uses of rhetoric and persuasion....
  
  lists several faults:

"In the Introduction of a cause we must make sure that our style is temperate and that the words are in current use, so that the discourse seems unprepared. An Introduction is faulty if it can be applied as well to a number of causes; that is called a banal Introduction. Again, an Introduction which the adversary can use no less well is faulty, and that is called a common Introduction. That Introduction, again, is faulty which the opponent can turn to his own use against you. And again that is faulty which has been composed in too laboured a style, or is too long; and that which does not appear to have grown out of the cause itself in such a way to have an intimate connection with the Statement of Facts; and, finally, that which fails to make the hearer well disposed or receptive or attentive." (Rhetorica ad Herennium, I. vii, 11, trans. Harry Caplan, Loeb Classical Library, 1954.)

In short, the exordium was the portion of the discourse in which the orator would prepare the audience to hear his arguments in a favorable frame of mind.