Question about irony

So today my roommate (who is an international student) wanted to use the word irony. He said he would use when describing a scene when he's telling someone "good job", but he really means "that was bad". I immediately told him, that was not irony, and is either deception or sarcasm. I gave an example of an ironic situation: a man wants to become healthier so he eat carrots, only to find that he is allergic to carrots and it makes him sick. My roommate then proceeded to show me the definition in the dictionary, and I saw this:

i・ro・ny
1. The use of words to convey a meaning that is the opposite of its literal meaning: the irony of her reply, "How nice!" when I said I had to work all weekend.
http://dictionary.reference.com/browse/irony
It also said something like this in the Merriam Webster dictionary.

On top of this, it said that one of irony's synonyms was sarcasm. So my question is: Is sarcasm irony, and can you say that sarcasm is ironic? I always thought people who used it like that were incorrect, and now I'm unsure.

Your friend's example is irony. Irony is a rhetorical device when you say something and mean the opposite.

Your example isn't irony, but it's similar. Although nobody says anything (so nobody can use a rhetorical device), the man tries to do one thing and ends up with the other through a twist of fate. Such a situation is commonly called "irony of fate". It doesn't have anything to do with literal irony, but I hope you can see the similarity. The (common) misconception that 'irony = irony of fate' probably comes from this song (I'm not kidding!), which was a massive hit single and therefore stuck in the minds of a lot of people. A better word to use for such a situation would be "paradoxical".

Sarcasm isn't strictly synonymous with irony, but they overlap. Synonyms for sarcasm would be "ridicule" or "mockery". Since irony is often used for mockery (like in the example your friend gave), the two are often confused. They're not the same however and sarcasm isn't strictly ironic. I can't come up with an example of irony that isn't sarcastic (if you include self-mockery in the definition), but there might be some.

Hm...thanks for the response SpiritCrusher. But now I have a couple more questions.

1. I noticed a lot of the examples in that song, as one my English teachers pointed out, aren't (to my knowledge) ironic, even by 'irony of fate'. Such as "rain on your wedding day". What separates my example from the one quoted from the song is that there is no reason for the reader/listener to believe that it will be sunny on one's wedding day, while in my example there is an understanding or belief that eating carrots is healthy for you. Given this, would the quote from the song actually be considered "irony of fate"?

2. Just for another example of what I thought irony was, this is an example of what I believed to be almost double layered irony:
"isn't it ironic that I'm the only one who can use the word 'ironic' in a sentence correctly?"
This sentence in itself is not ironic. However, the fact that I used the word 'ironic' incorrectly while stating that I was the only one who could use it correctly makes the sentence in fact ironic. Would this also be considered "irony of fate"?

EDIT: There is also this definition of irony:
5. An outcome of events contrary to what was, or might have been, expected.

Ultimate said:

1. I noticed a lot of the examples in that song, as one my English teachers pointed out, aren't (to my knowledge) ironic, even by 'irony of fate'. Such as "rain on your wedding day". What separates my example from the one quoted from the song is that there is no reason for the reader/listener to believe that it will be sunny on one's wedding day, while in my example there is an understanding or belief that eating carrots is healthy for you. Given this, would the quote from the song actually be considered "irony of fate"?

Click to expand...

Nope, "rain on your wedding day" doesn't have anything to do with irony of fate. It just so happens that the lyrics are really really stupid and Alanis Morissette has no idea what she's talking about.

Ultimate said:

Would this also be considered "irony of fate"?

Click to expand...

In a way. The thing with definitions is that you don't need to follow them to the letter.

Ultimate said:

There is also this definition of irony:
5. An outcome of events contrary to what was, or might have been, expected.

Click to expand...

http://tvtropes.org/pmwiki/pmwiki.php/Main/Retcon