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

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.

For the cultured

Irony is given by the simultaneous presence of two meanings, a deceiving one on the surface and a true one in depth, which may or may not be accessible to all.

A very common example of irony is that of saying something to convey the opposite of its face value: for instance, when something bad happens, one might say:

Oh, that's just great!

However, it is not necessary that the meanings be exact opposites. Here is a more sophisticated example of irony:

There are only 10 types of people in the world: those who understand binary, and those who don't.

In Socratic irony, Socrates feigned ignorance and asked questions to guide his disciples to a deeper understanding. Here the surface meaning is that of an apparent genuine inquiry, while the understanding that motivates it becomes clear to the disciple only at the end.

In the so-called irony of fate, due to their limited knowledge of the circumstances (or even to their lack of knowledge of the future) mortals act in a way that seems reasonable at first, but ends up having negative consequences in the end: here it is only Fate itself that knows the truth about people's destiny, and when they realize it it is too late.

Sarcasm is the use of wit or ridicule to taunt, mock or contemptuously strike at a target. Irony is a common tool for sarcasm, but it is by no means the only one. Often, a mere laugh can be sarcastic, without any use of words.

For the uncultured

Sarcasm: cheap irony.

Irony: when something goes contrary to expectations, and this is considered humorous or otherwise notable. Like rain on your wedding day.

On etymology under the entry for humor you will find a table that classifies humor and lists several qualities for both terms (from H.W. Fowler, "Modern English Usage," 1926):

• In terms of motive and aim, sarcasm aims to inflict pain, while irony aims for exclusiveness.

• For the province, sarcasm deals with faults and foibles, while the province of irony is statement of the facts.

• Regarding methods and means, sarcasm uses inversion, irony mystification.

• For the audience sarcasm is perceived by the victim and bystanders, while irony is intended for inner circle.

This is illustrated from the point of humor and, I believe, tries not to define, but to illustrate characteristic attributes for each category.

From the point of rhetorics, sarcasmus is figure of irony - this contradicts definitions of irony that emphasizes the intention of sarcasm and state that irony is not intentionally hurtful (as defining difference between two terms; although you will find dictionary entries that define sarcasm as "bitter irony").

Then, on top of all this, lot of people especially on the internet, will not make a distinction and use them interchangeably, so looking for proper examples is not easy.

Pragmatically speaking, I call something that was said sarcastic if I can call it mean (intention). I use it for spoken word or written comments, sometimes writings.

With irony it is not limited to something said or written, it can also describe situations which are not result of someone's intentions.

The difference is there in the definitions you found.

Leech, a famous linguist who sometimes appeared in my studies, while talking about Linguistics matters that I won't explain in depth, said something about irony (he mentioned it making a comparison with "banter", but we will leave this aside for the sake of this question); he said that it "is an apparently friendly way of being offensive".

Irony is used to convey, usually, the opposite meaning of the actual things you say, but its purpose is not intended to hurt the other person.

Sarcasm, while still keeping the "characteristic" that you mean the opposite of what you say, unlike irony it is used to hurt the other person. It might interest you to know that the etymology of sarcasm is this one:

ad. late L. sarcasm-us, a. late Gr. σαρκασµός, f. σαρκάζειν to tear flesh, gnash the teeth, speak bitterly, f. σαρκ-, σάρξ flesh.]