I would like to know what exactly makes matching markets different from regular markets. If you could illustrate with examples to help me understand better, that would be nice. Thanks in advance.
The deviance from a "normal" market boils down to how that market clears. Matching markets are markets in which prices alone don't clear the market, as thought in mainstream economic theory, and so the market is clear using the help of other institutions.
Take it from Alvin Roth's informative EconTalk interview:
"...labor markets are like these--they are among the markets that I've studied. And in labor markets, as in other matching markets, you can't just choose what you want. You also have to be chosen. So, labor markets have all these other institutions about applications and interviews and offers and acceptances. And sometimes those are used for competitive purposes in various ways. So, for example, right now if you knew someone who was just graduating this month from a prestigious law school and was going to take his first job as a clerk for an appellate judge, there's a good chance he would have gotten that job two years ago, when he was a beginning second-year law student. And he would have gotten that job with an exploding offer--which meant that he had to say yes or no when the offer came, before he could find out what other offers might be coming...And so that's a market that doesn't work the way we normally think a well-functioning market should work, where you can consider lots of alternatives. But it's one of the ways that judges compete with each other. And that's a market where the judges can't compete on price because Congress sets the wage for clerks. But it turns out that other markets work that way, too. When private equity firms are trying to hire young investment bankers, they do the very same thing. When law firms which pay salaries very freely, try to hire new associates, they do the same thing--often with competition through summer associateships while people are in law school. So that new lawyers in general will often, as they go to their jobs after finishing three years of law school, have been committed to those jobs for years."
In some other instances, consider trying to work for Stackexchange, where you can't just choose to work here: you have to be hired. Also this company doesn't actually try to lower it's price in order to clear the market like with commodities, but it will set a wage wide enough to be attractive to many people, in which it gets the opportunity to choose whomever it believes to be the best fit for their company. Or if we want to reverse the situation where you're a well sought after worker (perhaps a sports player, or a consultant) firms will express their strong interest in hiring you, but you can't just be hired because you have the opportunity to pick which of the employers you want to work for. I hope this helps.
Oh, and Consider Roth's book on the subject matter. Cheers.