A Derivative is any financial instrument which derives its value from some underlying security or other reference value. The security or reference in question can take many forms - e.g. stocks, shares, commodities, indexes, weather patterns and other derivatives. A simple example would be a call option on wheat. Wheat is the underlying commodity and the derivative is the call option. The underlying itself can be a derivative (e.g. options on options). The range of types of derivatives is huge. 1973, when the Black-Scholes pricing framework started to be used, is seen as the birthdate of the explosion of modern derivative trading.
Examples of vanilla options include calls, puts, caps, floors and forwards. Vanilla is a term which can be understood in a few ways. The most obvious is that the derivative has simple features, like a singular payoff, no-early exercise and so on. The other criterion is that the option can be priced in a straightforward manner.
Examples would include barriers, American options, rainbows, lookbacks, cliquet and baskets. Some people may describe some of these options as vanilla, although each of them provide unique challenges in quantitative analysis, making them non-trivial.
Derivatives with no optionality
Not all derivatives are options. The classic example is the rate swap, which is a bond derivative.
Size of the derivative market
The market for derivatives is of gigantic proportions. For example the notional value of all OTC at the end of 2006 was estimated to be $415 trillion, with a daily turnover of $2544 billion.