It happens. We’re human. Sometimes we make mistakes. Airlines are run by human beings, and sometimes they make mistakes too. Even the people who enter fares into computer systems.
Mistake fares are not always honored by airlines, but whether or not they do depends on the situation and how much leverage they have. For example, airlines have very wide latitude when it comes to administering their frequent flier program. In one case, United incorrectly published award seats at very low rates on their Web site. People who took advantage were contacted by United and had the choice to cancel the ticket and have their mileage returned, or to “buy up” to the correct redemption rate. United was fully within their rights to do this under the rules of the Mileage Plus program, a contract to which Mileage Plus members must agree to participate in the program.
Another case involving United involved cash fares. These fares were a really good deal: $0 to many destinations (including Hawaii!). While the error only lasted for about 15 minutes, hundreds of tickets were sold for only the cost of taxes. United ultimately decided to honor the tickets. They collected taxes and airport fees and they billed people’s credit cards. There is significant precedent in the United States that if you take someone’s money and promise to provide a service, you have created a contract. In the US, wishing you hadn’t signed a contract isn’t grounds to get out of it.
How do you find mistake fares? Look for them yourself – you can set up fare alerts on many travel sites, so set them up for your favorite routes. I personally use Twitter extensively. You can follow @TheFlightDeal, which I think is the single most authoritative source of mistake fares on the Internet.