Irony, at its root, is defined by the presence of two meanings.
Verbal irony has a lot in common with sarcasm in that what is said intends to be intepreted in a way contrary to the common meaning of the words used.
Ironic statements (verbal irony) often convey a meaning exactly opposite from their literal meaning.
Some examples of verbal irony might include Marc Antony's speech in Julius Caesar, where he states repeatedly that Brutus and Cassius are "honorable men". These men have just committed a murder. They've just stabbed a man in the back. The phrase then ("honorable men") is ironic because it implies an opposite meaning and underlines the corruption and immorality of the men being described.
Verbal irony can also relate to language with double meaning. For instance, imagine there were a situation on stage similar to that in Julius Caesar where a person had been attacked with a knife and a witty remark was made afterward. If another character were to say, "Ah. Your words cut to the quick!" this would be an example of verbal irony.