The word 'ironic' is one that I avoided using for a long time because I was very aware of the tendency for it to be misused, and I didn't feel I could reliably use it correctly myself. However, as I've gotten older, I've mixed it in more often and have caught myself misusing it just as I feared.
First, I don't think I was paying enough attention in high school English class when we covered irony, and thus, maybe I didn't fully grasp its meaning. Second, I think I just used it the way I'd heard others use it so often in the past—wrongly.
An example of this would be to say, "Ironically, the hapless Toronto Maple Leafs are playing their last game of the year on April Fool's Day." That is not ironic. You could say 'fittingly', 'appropriately', or maybe even 'coincidentally', but not 'ironically'.
The Huffington Post has an article on this exact topic—I've found tons—from about a year ago, where they cite Alanis Morrissette as one of the most widely known and classic examples of misusing 'ironic':
Sure, her song “Ironic” notoriously misuses the word (around 10 times) and repeatedly insists that rain on your wedding day is ironic rather than simply a bit of bad luck. But while many of us have poked fun at Morissette’s poor diction, her use of ironic to describe coincidences or annoying events is reflective of how many of us casually misuse the term. She may not be a particularly egregious offender at all, just more prominent. Isn’t it ironic? Actually, no, not really. But it’s easy to misuse the term in a world where so few of us know what the correct meaning is.
It's worth a read. They go on to explain several different types of irony, and how to use them correctly.
I found another site, called thatsnotironic.com, and it's a short and sweet explanation of the difference between irony and coincidence, including some hilarious examples from George Carlin, including this one:
For instance: a diabetic, on his way to buy insulin, is killed by a runaway truck. He is the victim of an accident. If the truck was delivering sugar, he is the victim of an oddly poetic coincidence. But if the truck was delivering insulin, ah! Then he is the victim of an irony.
In the end, this one is really just more fun to pick on than it is an egregious abuse of the English language. It's an extremely common error, and I wouldn't worry too much about it if I were you. The only people who care about this stuff are me, and a handful of other pedantic nerds out there.
Here's a video of George Carlin picking on some of his English language pet peeves, including 'irony' (at about 1:45):