Here is a "real world" example not in the sense that somebody happened to come across it in research, but in the sense that it uses everyday concepts without many statistic-specific terms. Maybe this way of saying it will be more helpful for some people whose training is in other fields.

Imagine that you have a database with data about patients with a rare disease. You are a medical graduate student and want to see if you can recognize risk factors for this disease. There have been 8 cases of the disease in this hospital, and you have recorded 100 random pieces of information about them: age, race, birth order, have they had measles as a child, whatever. You also have recorded the data for 8 patients without this disease.

You decide to use the following heuristic for risk factors: if a factor takes a given value in more than one of your diseased patients, but in 0 of your controls, you will consider it a risk factor. (In real life, you'd use a better method, but I want to keep it simple). You find out that 6 of your patients are vegetarians (but none of the controls is vegetarian), 3 have Swedish ancestors, and two of them have a stuttering speech impairment. Out of the other 97 factors, there is nothing which occurs in more than one patient, but is not present among the controls.

Years later, somebody else takes interest in this orphan disease and replicates your research. Because he works at a larger hospital, which has a data-sharing cooperation with other hospitals, he can use data about 106 cases, as opposed to your 8 cases. And he finds out that the prevalence of stutterers is the same in the patient group and the control group; stuttering is not a risk factor.

What happened here is that your small group had 25% stutterers by random chance. Your heuristic had no way of knowing if this is medically relevant or not. You gave it criteria to decide when you consider a pattern in the data "interesting" enough to be included in the model, and according to these criteria, the stuttering was interesting enough.

Your model has been overfitted, because it mistakenly included a parameter which is not really relevant in the real world. It fits your sample - the 8 patients + 8 controls - very well, but it does not fit the real world data. When a model describes your sample better than it describes reality, it's called overfitted.

Had you chosen a threshold of 3 out of 8 patients having a feature, it wouldn't have happened - but you'd had a higher chance to miss something actually interesting. Especially in medicine, where many diseases only happen in a small fraction of people exhibiting in risk factor, that's a hard trade-off to make. And there are methods to avoid it (basically, compare to a second sample and see if the explaining power stays the same or falls), but this is a topic for another question.