Some people also use the term to describe the aroma and essense of freshly prepared food cooked in or even out of a wok. But I don't think these people are right. A freshly prepared dish with fresh ingredients cannot have the breath of a wok. I understand that the amazing essence fresh ingredients prepared fresh is something just as hard to describe just as wok hay is hard to describe, but they are talking about something entirely different, mixed up with the adjacent properties of wok hay.
The other definition of wok hay is simply its flavor. Probably an average man with an average palate can quickly learn to identify a dish with wok hay when presented with 2 identical dishes (one with wok hay, and one without). But where the challenge is when one tries to describe the flavor. Why? The flavors and aromas are elusive. First, the flavor is different from dish to dish. Second, its more about the method/process than the ingredients, even though the ingredients are integral. Third, its invisible and we make the mistake on trying to identify the ingredient rather than recognizing the method/process. Fourth, people generally don't cook on a wok at home, and even if they did, they don't have the right equipment to generate the heat required so they don't know where the flavor comes from.
So how do you achive wok hay? You need a non teflon surface (a wok is almost impossible to do without), heated enough where the oil would start smoking almost immediately. And while its smoking, you should throw in your stir fry items. Then let the smoke catch fire while you're tossing the things inside. This process marries to elusive flavor to your seasoning.
Here's the proof to my logic. Say you have every equipment, preparation, ingredient, technique, and skill required to cook on a wok. On one wok, you cook at high enough heat for the oil to smoke, and when it smokes you let it catch fire. On the second wok, you cook just below the smoke temperature of the oil. At the end of cooking, food from both woks may look identical, but you can say that food from the first wok has wok hay, and food from the second wok hay has absolutely no wok hay at all.
The second part of my proof is that if you've ever accidentally cooked on a pan where the oil started to smoke, and some of the splattering oil and smoke accidentally caught on fire, and you ate the food anyway. You can argue that there was something extra in there where the flavor was kinda nipping at the heels of flavors associated with wok hay. You can even say it had some wok hay, whereas the food from the second wok mentioned above has no wok hay.
So there you have it. I hope I broke wok hay down so its more tangible. Unfortunately any home kitchen stove won't be able to produce enough heat to cook effectively on a wok. It may even be dangerous too, and I'm pretty sure the smoke detectors would go off. So even though you won't be able to achive wok hay at home after reading this article, you can probably try to nip at the heels of wok hay. Another thing to point out is that it is pretty common knowledge that oil that has reached its smoke point is supposedly very unhealthy with unspeakable amounts of carcinogens and free radicals. Its a little ironic because wok hay is generally associated with freshness and life. Wok hay is translated as "wok breath". Why is it that the better the food tastes, the worse the food is for you? It seems that wok hay is no exception to the rule. I'm sure I'll get no arguments when I say a life without blissful culinary pleasure is a life not worth living. So enjoy your carcinogens and free radicals in moderation.