A friend asked me to write this. If you're like her, you've scratched your head when a recipe specifies to use a non-reactive bowl, pan, skillet, etc. I'll begin by saying that for the most part, the issue of reactivity is becoming less and less of a concern as nearly all the cookware you can buy in stores today is non-reactive in nature. Unless you are using a lot of hand-me-down or vintage items, you need not think twice about reactivity.
But, rewind the clock a couple decades and we lived in a time when many mixing bowls, pots and skillets were made from "softer" metals like aluminum, tin, steel and copper. When a food with high salt content, or a highly-acidic nature (e.g., tomatoes, citrus) is introduced into a vessel made of soft metal, a chemical reaction takes place whereby little bits of metal get released, in turn leading the food to take on a metallic taste (and, in the case of paler foods, a bit of discoloration as well.)
To help avoid reactivity mishaps, it became common practice for recipe writers to specify when the use of a non-reactive bowl or cooking surface was necessary. And today, even though most of our cookware is made of non-reactive material - stainless steel, ceramic, enamel-coated cast iron, etc. - this remains a common practice in cookbook, food blogs, etc.
If you have older items in your kitchen, and are unsure what they are made of, it is best to avoid using them for recipes that call for non-reactive cookware.
One last thing - I was curious to know if cast iron is considered reactive, so I did some digging and found that a well-seasoned cast iron pan is unlikely to produce reactive effects. A less-seasoned (or unseasoned) cast iron pan may cause issues, although to a lesser extent than a super-soft metal like copper.