Having worked in investment banking for a while here is a short intro to some of the financial products that can be traded by investment banks.
If (like me) you like to really understand what something physically *is* then a derivative is a contract, an agreement signed between 2 or more parties. The contract will specify some sort of underlying asset and what to do with it at what price at some point in the future. Common, standardised, derivatives are referred to as vanilla and can be traded on an exchange, more complex ones are exotic and would be traded over the counter. Although valuing a derivative can be complex they do have a value so can be traded. Futures, options and swaps are types of derivatives.
A future is a standardised agreement to buy or sell the underlying asset for a specific price at a specific point in the future. Because futures can be traded on a futures exchange the buyer and seller don't have to know one another. Futures are often used for agricultural commodities and natural resources. Non-standard futures are called forwards.
An option is similar to a future but conveys the right, not the obligation, to buy or sell the underlying asset for a specific price at a specific point in the future.
A swap is an agreement to exchange something for a specific period of time. The 'something' is usually interest on a notional amount. One cash flow is usually fixed (i.e. 5%) with the other variable (BoE Base + 2%). The different cash flows are referred to as the legs of the swap.