What word is that? Business?
Keep in mind that we are referring to children (under the age of 18.) These sorts of transactions are often the first steps into the realization that you can grow value (which could be physical or virtual items) through transactions with others (this could be smart trading or selling something you created.) These first steps could be powerful lessons which may shape their future.
If I sell a baseball card and I make a profit, then that profit would likely be recorded on a Schedule C (Form 1040), Profit or Loss From Business.
http://www.irs.gov/uac/Schedule-C-(Form-1040),-Profit-or-Los...
Just by selling that card, you are a sole proprietorship.
If you get creative, you can think of many different ways in which children participate in business activity.
ETA: Okay, I get it. My examples aren't of people starting companies. A kid selling baseball cards might be a business, but not a company (company is a group, more than one.) I wonder if that's what he actually meant though. Can a minor even hire employees?