meaning - What are the similarities and differences between "irony" and "sarcasm"?

This seems to be one the long-standing arguments between people on the internet. When is something "irony" and when is it "sarcasm"? And can a quip be both at the same time?

Dictionary definitions don't seem to help much:

irony — the expression of one's meaning by using language that normally signifies the opposite, typically for humorous or emphatic effect

sarcasm — the use of irony to mock or convey contempt

In most of the arguments I hear about the words, neither of these definitions seem to match the definitions proposed by anyone.

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Per Rhodri's request, I typically see definitions such as these offered instead of the ones I found in my dictionary:

sarcasm — using a sentence to convey its exact opposite meaning

irony — a circumstance that involves one's intent or actions backfiring and bringing about the opposite of what was intended, usually through humorous or coincidental means

Note that this just my construction. People have offered all sorts of other working definitions but the big difference is that "irony" involves a circumstance instead of an expression.

Answer

Defining the word "sarcasm" is fairly straightforward. It usually means the expression of a sentiment whose opposite is meant. For example:

Hooray! I have a headcold.

Irony, however, eludes a simple explanation, and there tend to be disagreements about its meaning between UK and US speakers. Irony is found in the contrast between expected, or ordinary, outcomes and what actually happens. The greater the distance, the greater the irony.

For example: In this video, the police officer is giving a lecture on firearms safety, and shoots himself in the leg: http://www.youtube.com/watch?v=am-Qdx6vky0

That's an example of situational irony. And note that irony need not be funny, although it often is.