It is an instrument with certain rights/obligations and that "derives" its value from the underlying security (whether the security is debt, equity etc).

They are important given that we live in an uncertain in the world. A futures contract will provide you with a guaranteed selling or buying price at some time in the future. Other instruments such as an option contract gives you flexibility given that it allows you to reserve a buying or selling price for the future if your thinking of possibly buying or selling then - effectively it gives you a price floor for what you will be selling at and a price ceiling for what you will be buying at.

There are many many different types of derivatives out there with different functions.

Just to add on to Dorian's answer:

Derivatives are important as they can guarantee outcomes in the future by paying an amount today (much like insurance). Large businesses have many uncertainties but some risks are completely unwanted in their opinion and they can purchase derivatives to eliminate these, this is commonly known as hedging.

They can also act as leverage and allows small time investors to trade many times their equity (I think you pay a margin of circa 10% (I'm not sure of this could be more or less) of the contract value, therefore you can potentially trade 100k worth of value using just 10k however you still eat losses so it is obviously very high risk)

Just a quick distinction of the common derivatives and their potential use:

futures: Basically an obligation to buy or sell a certain amount of an asset at a certain price in the future which is determined today. * an apple farmer is thinking of expansion in the very near future and wants to be very sure that his profits this quarter at least reaches a certain threshold, therefore he can purchase a futures contract on apple prices.

options: Basically the right but not the obligation to buy or sell a certain amount of an asset at a certain price for the option holder, the option issuer's outcome basically depends on the buyer's reaction. There are two basic types of options, call (right to buy) or put (right to sell) options.

call options

You buy a call option contract which allows you to purchase 500 shares for Telstra at $3 ea anytime for the next 3 months for a cost of say, 8 cents per share so 500 x 0.08= 40 (lets ignore commissions for this exercise). The seller of the option gains 40 dollars once the deal takes place. If the stock price of telstra rises to say $3.50 then the buyer of the option could exercise or use the call option contract and purchase 500 shares at $3.00 from the issuer and immediately sell them for $3.50 each on the market. If the price of telstra is <$3.00 the buyer would basically never exercise the option because he could just get it on the market instead.

put options

Basically a reverse of the call option, giving you the right to sell at a certain price instead. Lets assume the exercise is still $3.00 ea for 500 shares of Telstra. The purchaser of the stock option could now exercise the option if the price of Telstra dropped to $2.50 by immediately purchasing 500 shares of telstra and selling them at $3.00 to the issuer. If the share price was over $3.00 then one would just not exercise the option. The maximum risk for the purchaser of the option basically is just the option price.

swaps: Basically an agreement where parties counterexchange each others benefits. This is normally driven by comparative advantage, therefore both parties are better off. Swaps can take quite long to explain clearly without diagrams so I suggest you watch this video on interest rate swaps thanks to youtube: http://www.youtube.com/watch?v=vzRnT-tpfmQ If you are still a little unclear I will be happy to reply when I can. Hope this helps!

If you are seriously interested and looking for an introductory text that keeps the technical level at the very minimum, then I can only recommend John Hull's standard textbook "Options, Futures and other Derivatives" (also used in FINS3635).