Irony.

[Ostinato]

You think you get it, but you really don't.

Sarcasm is not irony.
Coincidences are not ironic.
And puns are definitely not ironic.

When in doubt, just avoid that label altogether. You're not impressing anyone anyway.

[MrsSallyBakura]

Thanks for the vocabulary lesson.

[JesusRocks]

Also,

Irony works differently verbally, or in normal speech, than it does in literary form. Sarcasm is regarded as a form of irony.

I found this, rather useful, definition of Irony from a website for literary terms:

IRONY: Cicero referred to irony as "saying one thing and meaning another." Irony comes in many forms.
Verbal irony (also called sarcasm) is a trope in which a speaker makes a statement in which its actual meaning differs sharply from the meaning that the words ostensibly express. Often this sort of irony is plainly sarcastic in the eyes of the reader, but the characters listening in the story may not realize the speaker's sarcasm as quickly as the readers do.
Dramatic irony (the most important type for literature) involves a situation in a narrative in which the reader knows something about present or future circumstances that the character does not know. In that situation, the character acts in a way we recognize to be grossly inappropriate to the actual circumstances, or the character expects the opposite of what the reader knows that fate holds in store, or the character anticipates a particular outcome that unfolds itself in an unintentional way. Probably the most famous example of dramatic irony is the situation facing Oedipus in the play Oedipus Rex.
Situational irony (also called cosmic irony) is a trope in which accidental events occur that seem oddly appropriate, such as the poetic justice of a pickpocket getting his own pocket picked. However, both the victim and the audience are simultaneously aware of the situation in situational irony. Probably the most famous example of situational irony is Jonathan Swift's A Modest Proposal, in which Swift "recommends" that English landlords take up the habit of eating Irish babies as a food staple.

It's from this website: http://web.cn.edu/kwheeler/lit_terms_I.html
And it's not just some person making it up, if you look to the bottom of the page, there are the many texts the website creator has used in defining the literary terms on the site.

I also found this from Cracked.com, one of the Comments posted to an article references the Oxford Dictionary of Literary Terms. Irony is defined thus:

"a subtly humorous perception of inconsistency, in which an apparently straightforward statement or event is undermined by its context so as to give it a very different significance."

Some coincidences can also be ironic - but most coincidences are not.
Puns... yeah you're right with that... puns aren't ironic

And, this is how you give a legitimate vocabulary lesson - with references.
I'm now wondering if there's any chance this thread could be revived... eh probably not... it's a flame war waiting to happen.