What it is:

A derivative is a financial contract with a value that is derived from an underlying asset. Derivatives have no direct value in and of themselves -- their value is based on the expected future price movements of their underlying asset. 

How it works/Example:

Derivatives are often used as an instrument to hedge risk for one party of a contract, while offering the potential for high returns for the other party. Derivatives have been created to mitigate a remarkable number of risks: fluctuations in stock, bond, commodity, and index prices; changes in foreign exchange rates; changes in interest rates; and weather events, to name a few.

One of the most commonly used derivatives is the option. Let's look at an example:

Say Company XYZ is involved in the production of pre-packaged foods. They are a large consumer of flour and other commodities, which are subject to volatile price movements.

In order for the company to assure any kind of consistency with their product and meet their bottom-line objectives, they need to be able to purchase commodities at a predictable and market-friendly rate.  In order to do this, company XYZ would enter into an options contract with farmers or wheat producers to buy a certain amount of their crop at a certain price during an agreed upon period of time. If the price of wheat, for whatever reason, goes above the threshold, then Company XYZ can exercise the option and purchase the asset at the strike price. Company XYZ pays a premium for this privilege, but receives protection in return for one of their most important input costs. If XYZ decides not to exercise its option, the producer is free to sell the asset at market value to any buyer. In the end, the partnership acts as a win-win for both parties: Company XYZ is guaranteed a competitive price for the commodity, while the producer is assured of a fair value for its goods.

In this example, the value of the option is "derived" from an underlying asset; in this case, a certain number of bushels of wheat.

Other common derivatives include futures, forwards and swaps.

Why it Matters:

As often is the case in trading, the more risk you undertake the more reward you stand to gain. Derivatives can be used on both sides of the equation, to either reduce risk or assume risk with the possibility of a commensurate reward.

This is where derivatives have received such notoriety as of late: in the dark art of speculating through derivatives. Speculators who enter into a derivative contract are essentially betting that the future price of the asset will be substantially different from the expected price held by the other member of the contract. They operate under the assumption that the party seeking insurance has it wrong in regard to the future market price, and look to profit from the error.
Contrary to popular opinion, though, derivatives are not inherently bad. In fact, they are a necessity for many companies to ensure profits in volatile markets or provide mitigated risk for everyday investors looking for investment insurance.

Best execution refers to the imperative that a broker, market maker, or other agent acting on behalf of an investor is obligated to execute the investor's order in a way that is most advantageous to the investor rather than the agent.