A stock option is a good example of a derivative.
You might purchase a call option on XYZ company at $40/share that expires in August. That would give you the right to purchase 100 shares of XYZ at $40/share. Now let's say XYZ is at $35 today. You might pay $4/share for the option. If XYZ goes up to $40, you lose. If XYZ goes up to $50 before August, you could exercise your option to buy shares at $40, then immediately sell them at $50 (the market price) and make a quick $10/share profit. Subtract the $4/share the option cost, and your net profit is $6/share. That doesn't sound like much, but it's a 150% profit on what you invested in the option.
Now if you owned the stock outright at today's price of $35, you would still make money from the stock going up $15/share, but it's only a 43% profit.
The point is that an option can make you a much higher percentage profit, but your risk is high. If the stock doesn't go up, your option expires worthless and you lose everything. This is oversimplified a bit but it illustrates the idea.
The idea behind it is that small changes in price can generate large profits (or large losses). It increases your leverage on the money you invested, just like a down payment on a house. If you put $10,000 down on a $100,000 house, and sell it a year later for $110,000, you doubled your money approximately. The $10,000 you invested grew by another $10,000.