Think of derivatives as paper contracts designed for betting that an asset price will go a certain way.
Simple example - I bet you some money that a certain asset will be worth at least $100 tomorrow. If tomorrow it is $120 then I make a profit and you make a loss but if it turns out to be $90 then I make a loss and you make a profit. I don't physically own the asset but I made a bet that was 'derived' from the price of the asset (hence the term 'derivative').
Another (more realistic) example below to illustrate what derivatives are and how they work:
Suppose I'm a company who uses oil in my products and am worried about the volatility of oil prices in the next year or so. If I simply buy the oil directly off the market this time next year, I risk buying the oil too expensively than I would be comfortable with.
However, what I can do instead is negotiate with my 'wholesaler' that I want to lock in the price at say $70 a barrel when I buy the oil this time next year. If this is agreed upon, then a contract is written up (known as a forward contract) where I am guaranteed that price next year (think of it like a term deposit).
If this time next year, the oil price turns out to be $90 a barrel then that contract worked in my favour BUT if it was $50 a barrel instead then I lose out (as I could have gotten it cheaper at $50 directly from market rather than the $70 in my contract). However, the benefit of the contract was that it removed the uncertainty of not knowing what price I will pay for the oil in the future.
However, the contract itself is also worth money. If the oil price is likely to tank below $70 then then a contract locking in a more expensive price may be worthless to someone else - but if the oil price soars above $70 instead then someone may be willing to pay some good money for the contract due to cheaper price locked in the contract (FYI there is a whole branch of mathematics dedicated to pricing derivatives).
Hence, I can sell that contract (before it matures of course) to someone else who is willing to buy it and I can potentially make a profit if the oil price works in my favour.
Now, if you imagine the many variations of this example (e.g. on other assets) and more complicated contract arrangements (e.g. futures, options, CDOs) on a large scale then you have an entire market trading these contracts. This is the derivatives market.