We make the majority of our cookware out of just a few basic materials: ceramics (including earthenware, stoneware, and glass), aluminum, copper, iron, steel, and stainless steel. Ideally, we want our cookware to conduct heat efficiently and not cause discoloration or off-flavors in our food. Unfortunately, no single material can do both things.
Ceramics and stainless steel are considered non-reactive. While these don't conduct heat very well and tend to have 'hot spots,' they won't interfere with the chemical structure of the food in such a way that changes the look or edibility of our food. Their other big advantage is that once they're hot, they stay hot for quite some time!
Aluminum, copper, iron, and steel (not 'stainless') are all reactive. They conduct heat very efficiently, and therefore, do a great job of cooking our food evenly. However, these metals are reactive with acidic and alkaline foods. If you're cooking with ingredients like tomatoes or lemon juice, your food can take on a metallic flavor, especially if the cooking time is very long. Light colored foods, like eggs, can develop gray streaks.
Foods will also pick up chemical elements from reactive cookware, causing us to ingest metals like copper and iron. Our bodies process iron relatively easily, so using iron cookware regularly isn't a problem. Our bodies have a harder time eliminating copper. When copper cookware is used to occasionally whip egg whites or sautéd vegetables, the small amount we ingest isn't enough to harm us, but you definitely don't want to use copper for every day use.
To get the best of both worlds, manufactures try to find ways to combine elements. Adding a layer of copper to the bottom of a stainless steel pan or coating iron with enamel helps to heat the pan evenly while still protecting food from direct contact with the reactive metal. Unfortunately, these kinds of cookware don't come cheap and can be hard on the ol' wallet.
In brief, use non-reactive cookware whenever your dish contains acidic or alkaline ingredients. Cookware made with reactive metals is a good choice for boiling water, sautéing vegetables, or searing meat (though don't deglaze the pan with an acid!).
Hopefully this helps clear up some of the confusion!
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Related: Do You Use Nonstick Cookware?