This is a good idea for the teenager who's earned a nice chunk of change from a summer job and wants to see that money grow even more. By investing in the stock market, you learn the basics of how the market works, the risks and benefits of investing, and the identity of this "Dow Jones" character is that they're always blabbering about on the news.
You have to be over 18 to have your own brokerage account. So, start with your parents' investment accounts, adding your money to their portfolio of stocks, bonds, CDs. When your folks get their monthly statements, have them show you how the changing price of different investments affects the overall value of the portfolio. Your parents might even let you choose a specific stock to buy, and then you can follow news headlines about the company, its earnings and its stock price and see how these components work together. Later, they could open a custodial account for you where they make the investments on your behalf, but the proceeds must be spent on you.
Another strategy is mutual funds like the Monetta Young Investor Fund or the USAA First Start Growth Fund. The Monetta fund has a $100 minimum initial investment with automatic monthly investments as low as $25, plus it sends age-appropriate "financial literacy kits" with newsletters, games and other perks to investors as young as newborns.