A contract between market participants, the value of which is derived from the action of a good or service. It is based on the concept of leverage. The buyer pays the seller a relatively small amount for an agreement to purchase a good or service at a specified future date, or given a certain action in the future. The buyer then has control of an asset with a much higher value than the initial price paid.
Options or futures contracts are derivatives. That means, they're based on something else. An option contract might say "You have the option to buy this stock at this price on this date" but you can buy and sell the actual contract. Buying and selling the contract as opposed to buying and selling the stock is different. The contract is a derivative of the stock. It's existence is derived from the the stock's existence.
If you hear someone say the "derivative" market, they're probably talking about the market of buying and selling options contracts for stocks.