One of the most important things I learned from my Economics professors at UCLA was at a cocktail party at the late Sherwin Rosen's place in Rochester, New York. Labor economist Finis Welch was visiting from UCLA and I had left UCLA to be an assistant professor at the Rochester B-School the previous summer. How we got to this topic doesn't matter (although, believe it or not, I remember) but Welch used the term "asymmetric loss function" and said that if you didn't have one, you would miss half your flights.
I got the point immediately. The asymmetry is that a mistake in one direction (say, too much) has a much lower cost than a mistake in the other direction (in this case, too little.) Back to flights. If the loss to you of being 30 minutes too early for a flight is the same as the loss to you of being 30 minutes too late, that is, if you had a symmetric loss function, you would be late roughly as many times as you would be early. Of course, we're not. And the reason we're not is that the cost of being 30 minutes too late is typically a large multiple of the cost of being 30 minutes too early.
Once I had that concept, I started seeing all kinds of decisions through that lens. Should I pack a raincoat just in case it rains? Typically, the cost of having the excess weight is small relative to the cost of not having the raincoat if it does rain. So yes, I should. If I have a sore throat and want some lozenges to carry me through the day, should I carry 4 in my pocket (the number I expect to use) or 8? The cost of not having enough is substantially larger than the cost of the extra little pocket space and the slightly degraded wrappers on the unused lozenges. So I should carry 8.
Of course, I've ignored probabilities here. If I'm packing for a trip and just taking a carry-on, I might not take the raincoat if the weather forecast for where I'm going is less than a 10% probability of rain. So there are some implicit probabilities lurking in the background.
So let's have a fun game. (I think it's fun, at least). Give some actual examples from your life where you have an asymmetric loss function and the asymmetry is huge.