An egregious mistake, failure, or problem is not merely bad. It is so bad that it stands out; people cannot miss it. Newspapers, lawyers, and teachers reach for it when they want to name something that should never have happened: an egregious error, egregious abuse of power, an egregious breach of the rules. It is fairly formal. In the kitchen you would just say a terrible mistake. People sometimes stretch it for anything they strongly dislike, but the core is still this: wrong, and obvious. Though actually, the Latin it comes from pointed the other way. ēgregius was out of the flock in a good sense — outstanding, excellent. English kept the picture of standing apart from the crowd and, in normal modern use, almost always means standing out as badly wrong. A student will not meet the old praise meaning in everyday English.