A derivative is a financial instrument that does not constitute ownership, but a promise to convey ownership.
Examples are options and futures.
The most simple example is a call option on a stock. In standarized options sold through the Options Clearing House, there are supposed to be sufficient safeguards for the small investor against this. Before discussing derivatives, it's important to describe their basis. All derivatives are based on some underlying cash product. This is the buying and selling of foreign currency at the exchange rates that you see quoted on the news.