Debbie, y'all probably don't have grits up there in Canada. Basically, it's cornmeal mush, made from white corn. It used to be a breakfast thing, but creative chefs are beginning to use it for main dishes, as the Italians do with polenta. If you've ever eaten polenta, it's the same thing, only made from yellow corn.
Chantrells are an exotic mushroom, and jus means juice from the meat, so I'd say to use your imagination on that one. Perhaps he uses dried chantrells, soaks them in hot water, and uses the juice from them.
Mascarpone is an Italian cream cheese...I've never tried it, but I believe I've heard that it's sweeter than our cream cheese.
I'm guessing that the vinegar Iron Chef is using is some fancy kind of balsamic vinegar, which as actually somewhat sweet.