I am a fresh upstart derivatives trader, and I have broken even on all my trades so far, so I can help. 
A derivative is a contract based on another underlying asset.
Both the trading of assets and trading of derivatives based on them help establish prices for the assets.
eg. Gold is a commodity. An agreement to buy gold in three months at a certain price is a derivative on gold. Euro is a commodity. An agreement to have the right to buy 1 Euro for $100 next year is a derivative on Euros.
Derivatives are mainly futures (where you agree that you MUST buy or sell something at a specified price in future), options (where you agree that you MAY buy something at a specified price in future), options on futures, options on options, and swaps (where you “borrow” money from somebody, and “lend” it back to him, and then exchange interest on a purely notional principal).
Over the counter derivatives are demonized, because they are not exchange traded and are done without existing regulation or procedure other than direct phone calls. Some OTC derivatives like synthetic CDOs (explained here) are seen as being so funky that they might be outright gambling and a misuse of depositor’s money.