One of the most misused words in the entire English language.
There are several types of irony.
Socratic irony - When someone pretends to be naive about a certain subject, and uses his questions about it to point out a flaw in the established belief. This is often used on the TV show South Park, where the children often
ask questions about a situation until the folly in a parent's decision becomes clear.
Sarcasm - Understatement, mocking overstatement, or
heavy-handed irony (stating the
flat opposite of the
truth) where both parties are aware of the difference between what's said and what's actually happening.
Situational Irony - The irony that most people think of. A difference between what you expect to happen (in a story, for example) and what actually happens. Rain on your wedding
day would be a sort-of example, because a wedding
day is generally expected to be a perfect, happy
day. The good advice you didn't take, however, would NOT be irony, because that has nothing to do with what is expected and what isn't expected. A traffic
jam when you're already late wouldn't be irony either; there's no automatic expectation that traffic will be
fine, just because you happen to be late.
Irony of
Fate - The concept that the Gods, Fates, etc. are toying with humans for amusement by using irony. Beethoven's loss of hearing is a famous example; one would expect a composer to be able to hear his compositions, but
fate denied him that ability.
Tragic (
Dramatic) Irony - When the audience knows something that some of the characters
don't know in a play/movie/novel/whatever. For example, when the horror flick psycho is in the house and the homeowner just goes in without suspecting anything.
So, yeah,
Alanis was
wrong in a lot of her song, but there ARE some examples of irony in there - and a few that are kinda-sorta, but could be
better. The old man who buys a lottery ticket is one; it would be a better example if he won, and then died of a heart attack from the shock of winning.