There are different types of irony.
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Situational irony occurs when what happens is very different from or the opposite of what is expected or likely.

A new law is passed to help the elderly. The elderly are more burdened by this law.
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Two students have a test. The one who has studied for weeks has an anxiety attack, and she cannot remember anything. This student fails. The other student
forgets about the test. This student anticipates failure, but he does well.
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Five friends go to the mountains for a weekend. Four of them decide to learn to ski. All of them fall and slide, but they have a very good time. The man at the cabin trips over a rug and breaks his leg.
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Verbal irony occurs when someone says nearly the opposite of what he/she means/intends.  This is often sarcastic.  It might also be playful.  (to have a humorous effect)
Customers witness a store manager screaming at an elderly woman who begins to cry.  One says angrily, “Well, he is just charming.”  The other adds bitterly, “Yeah, he should be nominated for Man of the Year.”  
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Whenever someone gets news like the boss is in a bad mood or annoying relatives are coming to visit, and the person says, “Oh great!”
(This is similar to a writing style in which there is a noticeable difference between what is stated and what is intended.)
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Dramatic irony occurs when the audience knows something that the characters do not know. (like in a play)
The killer is actually in the room that the other characters are entering.  (The audience knows where the murderer is hiding, but none of the characters can see him.)