Since I became observant of the kosher laws while living in a largely non-Jewish area, I got a lot of practice in explaining kashrut in a quick-and-dirty way. This is what I came up with, and maybe it will be useful to you!
Eating kosher is essentially three levels of paying attention to what you eat.
1) What you’re actually eating. There are prohibited animals that can’t be eaten. Everyone knows pig, but I also don’t eat shellfish, shrimp, or bottom-feeder fish like catfish. There’s also shark, swordfish, and insects.
2) What you’re eating that item with. I don’t eat meat with dairy. This even includes meat with a side dish that has dairy, such as a steak eaten with a baked potato with butter and/or sour cream.
3) How you prepared it. Going back to what you eat the food with, I wouldn’t eat a steak cooked in butter. But more than that, I have two sets of kitchen supplies, one for meat and one for dairy, so that I don’t accidentally mix the two. Surprisingly, if you cook chicken soup in a pot one night and clean it well afterwards, and then boil water in it the next day, a professional chef could tell you what you cooked! Meat and dairy could be absorbed into the pot or plate or other utensil, so we separate them. That means I use pots, pans, silverware, and dishes that are designated as either meat or dairy. I don’t use other peoples’ dishes and pots because they’ve probably been used for both.
If the person is still interested and has more time, I spend a minute or two on each topic, fleshing it out. Usually, they jump right to questions about specific examples.
Have you ever tested the chicken soup thing? With what kind of pot?
I'll second Larry. I could see this happening with a non-stick pot or a cast-iron skillet, but not with a regular metal pot. Sorry.
I'm merely sharing what I've been told. I know nothing about the chemistry of cookware.
Sorry, but I think you've been told a bunch of nonsense by some kiruvniks re: the pot. Even from a halachic standpoint, it doesn't really make sense because then how could anything be kashered via hagalah?
I don't find this all that ridiculous because we're talking about a ben-yomo pot, one that has been used for hot food within 24 hours. As I said in the hypothetical, we're talking about a pot used the night before. Also, we are assuming a professional, not your run-of-the-mill taste tester. And whether it's ridiculous or not, it still illustrates the rule of halacha for someone totally unacquainted with the concepts of kashrut.
If cookware is less than 1/60th non-kosher then it is considered Kosher.
Anonymous: I think you mean food, not cookware.
However, I think that is far too technical to discuss in 30 seconds with someone who knows nothing about kashrut. I think throwing in that fact will make the conversation a lot longer than intended, confuse the listener, and not accomplish your point of (in most cases) helping them understand why you can't eat something they've just offered you. And of all technical details, why this one?? And once you've mentioned this one, why stop there? I think you missed the point of this post.