There are many kinds of derivatives, and this isn't the place to explain them all. Basically, a derivative is a financial instrument whose value depends on the performance of another financial instrument.
http://en.wikipedia.org/wiki/Derivative_(finance)
The simplest one is called an option. Say you think that the stock in company X, which is at $1, is going to rise by 20% this year, but you don't have the cash to buy a million shares. You can, for a much smaller sum, buy an option to purchase those shares a year from now at a fixed price, say $1.05. The option itself costs 5c (times 1 million is $50,000, an affordable sum of money compared to $1,000,000). So if you are correct, and the price goes to $1.20, you exercise your option. Now you've paid $1.05 for the share + 5c for the option = $1.10. Anyone will loan you the money to exercise the option because you can immediately turn around and sell the share at $1.20, meaning you have made 10c profit. For a million shares, that would be $100,000. If the stock didn't do as well as you hoped (ie didn't go past $1.10), then your option is worthless, you lost $50,000 - but at least you didn't tie up a whole million bucks in the process. This known as a 'call' option.
Alternatively, if you think the price is going to go down, you can buy an option at 5c to sell the stock at 95c. If the stock falls to 80c a year from now, you can exercise your option, buy the shares at 80c each + 5c for the option = 85c, and then sell the stock at 95c - again, you made 10c per share, times a million is a cool $100,000 profit on a $50,000 investment. Again, anyone will loan the money to complete the transaction because if you have the option and the company's share price has changed as you predicted, it's obvious that you're going to make a profit. This is a 'put' option. Again, if the stock didn't go down as much as you expected, the option is worthless, you tear it up and bemoan your loss.
It is quite possible to buy both call and put options, and make a profit whichever way the stock moves, as long as it moves by more than the value of your options. In this case, assuming the options cost 5c each, as long as the stock goes up or down by more than 10 cents you've made a profit. In this case you are making a bet that the price of that stock is going to be volatile and change significantly, even though yo are not sure in which direction. An example of when you might want to do that is if you heard the CEO was going to be fired, and you couldn't guess whether the new CEO was going to be competent or not, but you knew the shareholders were demanding major changes.
The key point here is that although you may have only invested $50,000 or $100,000, it is actually like making a deposit on a million dollar investment. A great many derivatives work along similar lines, so when you hear that the derivative market is worth a gazillion dollars, it doesn't mean there's a gazillion actual dollars in play, only that investors have made down payments on a gazillion dollars of POSSIBLE future transactions. A lot of these deals are going actually cancel each other out, since it's not possible for all such transactions to profitably take place.