Consider your audience. Teaching irony to a child is different than dealing with literary irony in a classroom of college students. Some people will have an easier time grasping sarcasm and irony than others. Simplify your discussions of irony if you are explaining to a child or if someone is struggling.
Understand the types of irony. Fundamentally, irony is always a break between what is said or done and what is meant or intended. This is true in terms of both rhetoric and literary irony. Verbal irony is common in our culture, most often heard in the form of sarcasm. Dramatic or tragic irony is present in theater. In this instance the audience is aware of the disparity, but the character is not. Situational irony is the discordance between the intended result and the actual result of an action.
Consider examples of verbal irony. In its broadest definition, verbal irony can include sarcasm, hyperbole, understatement and even rhetorical questions. Verbal irony is, however, always intentional. Many traditional phrases are an example of verbal irony, including "as clear as mud" or "as much fun as a root canal." It is necessary for the listener to have a basis of cultural knowledge in order to understand much verbal irony. If you do not know what a root canal is, you might not realize that a root canal is, in fact, not at all fun.
Explore dramatic and tragic irony. Tragic irony is found in many classic plays, particularly those of ancient Greece, and also in Shakespeare's tragedies. In tragic irony, the audience is aware of all of the facts of the situation, for instance that Oedipus has married his mother, but the characters are not. Dramatic irony shares some characteristics with tragic irony, but is less all encompassing. As a theatrical device, dramatic irony involves setting up the situation, the situation coming to a head as the characters become aware, and finally the ironic situation resolving in some way to conclude the story.
Look for examples of situational irony. This is the most modern use of a term that has been relevant for a very long time. Situational irony may refer to an unusual coincidence or unexpected happening that results in a surprise for those present or involved. While both verbal and dramatic irony are intentional, situational irony is not.