Ok, I'm going from memory here. But I'm pretty sure the whole derivatives thing started as a way to protect farmers.
If you were growing coffee, and wanted a guaranteed price at the end of the season, you could agree to sell X amount of coffee on a certain date for Y price per kg. That was a good deal for farmers and for coffee purchasers, as they could protect themselves from risks. This sort of derivative is called a future, and the trade you promise to make is mandatory. You've got to actually buy or sell the coffee at the end date. (Ok, generally no actual coffee changes hands, but you can't choose not to participate in the deal.)
This idea spread to currencies and stocks - you can agree to trade X amount of BT shares on a future date at a specific price. This can be useful to protect yourself against uncertainties in the stock market.
Options are a tiny bit more complicated than futures, because they are, as the name implies, optional. At least for one of the parties. An option gives you the right to purchase (or sell) a given underlyer at a specific price on a specific date. When that date comes, you can either decide to use the option (exercise it) or not, depending on the price of the underlying stock.
Options and futures cost less than the underlying shares, and have a greater risk, or chance of pay off. So you can take greater chances with your money, effectively. If you have strong feelings about the way the market will move, and your strong feelings are right, you can make a lot of money with options and futures. Everyone trading options and futures have sophisticated computer systems telling them how much a given option or future is worth, at a given point in time, based on current share price, historical share price, any pending dividends, any related currency shifts etc etc.