Investopedia defines a derivative financial instrument as a contract between two parties in which the contract's value is determined by the fluctuation in value of an underlying asset. The parties to the contract take opposite positions as to whether the underlying asset's value will rise or fall. The name "derivative" comes from the fact that the contract derives its value from the underlying asset.
Derivatives are usually used as hedging instruments, but they can also be used for speculation. Investopedia offers the example of inflation risk as a common reason investors hedge with derivative contracts. A common example is that of a European investor who buys U.S. stocks on a U.S. stock exchange. This investor is exposed to currency risk. He may see his profits from the U.S. stocks dwindle or be lost entirely if the value of his home country's currency moves in the wrong direction. To protect against this risk, this investor might buy a currency futures contract, a specific type of derivative financial instrument.
Investopedia cites many examples of underlying assets used to determine the values of derivative contracts. These include stocks, bonds, commodities, interest rates and market indexes. There are also derivatives based on non-financial instruments, such as the rainfall, temperature or number of sunny days in a certain region during a certain time period.