Irony is an implied discrepancy between what is said or done and what is meant or expected to happen.
The term is is commonly used in U.S. English to refer to a type of rhetoric whereby a statement or assertion is made with the intention to put across the opposite meaning. E.g. "Oh I really like your coat." In British English this is more commonly referred to as sarcasm. Whether a particular statement is construed as irony or sarcasm can often be a matter of opinion. The Sarcasm Society  offers the following distinction:
- Sarcasm can be illustrated by the speaker saying, "I appreciate your help.", when no help was provided to the speaker. However, when the speaker says, "It was a bit cold.", after losing a leg due to frost bite, this understatement can be construed as irony.
It might be useful to distinguish three kinds of irony :
Verbal irony, when an author says one thing and means something else. This is sometimes called 'Socratic Irony' and is used by teachers pretending to be foolish or ignorant, to expose the ignorance of another.
Dramatic irony, when an audience perceives something that a character in literature does not know. The effect can be tragic, such as in 'Romeo & Juliet', where the fate of the characters is sealed at the start of the play, or comedic, such as in 'Twelfth Night' where Malvolio is duped by Maria and Toby. Such irony can work in reverse, where some characters in drama understand their actions, but the audience is duped into thinking something else. Such is the situation in The Village, or The Sixth Sense.
Irony of situation, a discrepency between the expected result and actual results. An example of this definition of irony might be a boxer betting his life savings on a fight he expects to lose, and then accidentally winning the fight.
Other types of irony include:
Romantic irony, a narrative device whereby the author repeatedly reminds the audience of his presence in a written or performed work. Romantic irony can range from the subtle (the inclusion of a third chair in a scene with only two characters as a reminder of the author's invisible presence) to the overt (Lord Byron's repeated allusions to himself and his craft in "Don Juan.")
Cosmic irony is predicated upon the notion that the fates, the gods, or the universe are either capricious or ultimately indifferent to human suffering. Thus, irony results from the contrast between the pathos of mankind's struggles and their ultimate futility. Cosmic irony is a hallmark of the naturalist literary movement; the works of Thomas Hardy, in particular, invoke cosmic irony on a routine basis.
The current situation is further clouded by the fact that literature, cinema, fashion and behaviour can now be described as post ironic. This term is often used when a statement is made which is insulting, and which potentially could be true, is not meant as an insult. An example of this is being asked "Does my bum look big in this?", and replying: "It's looks bigger than an elephant's". This taken on face value could be seen to be insulting. However the implication here is that this latter statement is ironic, and no insult is intended. The problem with post irony is that it is often difficult to distinguish between fashion gaffs, insults and poor literature and attempts to be 'cool', 'subtle', and 'sophisticated'. 
- definition adapted from: http://www.tnellen.com/cybereng/lit_terms/irony.html
- For further clarification on this see the urban dictionary definition at http://www.urbandictionary.com/define.php?term=post-ironic