Derivatives are financial instruments that derive their value from an underlying investment.
Put simply, a derivative is a contract between two parties in place around the performance of, for example, a stock, currency or index. These contracts are often bespoke and are in many cases zero sum, which is to say that the result will be to the detriment of one party and the benefit of the other.
Here are a couple of examples:
Company 1 owns a stock but wishes to avoid the risk of movement in price of that stock, so it agrees to pass it to company 2 for a period of time. Company 2 wishes to benefit from a potential rise in price of the asset, so enters into a derivative contract with Company 1 for the difference in price of the stock over a period of time. Company 2 pays Company 1 interest over this period. The stock does not change hands and only the difference in price over the period one way, and the interest the other, is physically transferred between the parties.
This is called a contract for difference.
Company 1 is a farmer who wishes to sell pork bellies to Company 2 at a specific date in the future for a specific price. Company 1 is seeking to lock in the price it will be able to attain and avoid the risk that prices will reduce in the interim. Company 2 is betting that the price will rise over the period and they can purchase the commodity at a cheaper price than will potentially be available at that future date. They enter into a contract with those specific terms, with delivery at the date set.
This is called a commodity future.
There are many more types of derivative contracts, such as options, warrants, credit default swaps, currency futures and forwards. Some are similar to the above but others, such as warrants or options, for example, give a party the right but not obligation to exercise the contract.
Derivatives are widely used to hedge risks within a portfolio of assets, moving risks away from the company to another party who is willing to accept the risk speculatively for potential gain. The currency hedging article contains an example of this.