Irony --the expression of one's meaning by using language that normally signifies the opposite, typically for humorous or emphatic effect.
"Don't go overboard with the gratitude,” he rejoined with heavy irony.
synonyms: sarcasm, causticity, cynicism, mockery, satire, "that note of irony in her voice"
The only thing ironic about the Alanis Morissette song "Ironic" is that these are not examples of irony. They are actually coincidence or just bad luck. So, since you would think a song called "Ironic" would be full of ironies, but it is not. That makes the song ironic.
It is not ironic if it rains on your wedding day. It is just unfortunate. If you were paranoid that it was going to rain on your wedding so you decided to have to have the wedding inside, but then the pipe broke drenching everyone -- well, that's ironic.
George Carlin on Irony: Irony deals with opposites; it has nothing to do with coincidence. If two baseball players from the same hometown, on different teams, receive the same uniform number, it is not ironic. It is a coincidence. If Barry Bonds attains lifetime statistics identical to his father’s it will not be ironic. It will be a coincidence. Irony is "a state of affairs that is the reverse of what was to be expected; a result opposite to and in mockery of the appropriate result." For instance: