Foods cooked in reactive pots will often pick up a metallic flavor and sometimes turn funny colors, particularly very acidic or very alkaline foods. Whipped egg whites can develop gray streaks and tomato sauces start tasting tinny.
The chemical reactions that cause these things take a little while to happen, so you don't really need to worry if you're just doing a quick sauté or using aluminum bowls to prep ingredients. But if you're cooking a slow-simmered sauce or preparing something with very delicate flavors, then you might want to find a nonreactive pot to use. These are also the reasons why a recipe might specifically tell you to use a nonreactive pan.
As you can probably guess, nonreactive cookware doesn't react with food at all. You never have to worry about using it in the kitchen. The downside is that it's usually more expensive!
By the way, copper cookware is the exception to the rule. Copper is a reactive metal, but it's valued for things like whipping egg whites (makes the whites more stable) and quick cooking (because copper conducts heat very well). It's also pretty darn expensive!