What is a derivative?Easy

A contract whose value is read off something else. That is the whole definition, and everything difficult about derivatives is a consequence of it rather than an addition to it.

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The short answer: a derivative is a contract between two people whose value is derived from something else — a share price, an interest rate, a currency, a barrel of oil, even the weather. Nobody buys the thing. They agree in advance what will be paid depending on what the thing does.

Why anybody bothers

Three reasons, and only the first is what the instruments were built for:

To make a plan hold. An airline knows what it will pay for fuel next year; a company with dollar revenue knows what it will receive in euro. The gain or loss on the contract offsets the thing being planned around. See hedging.
To take a position without owning the thing. Cheaper, faster, and possible in the other direction — you can be short a market without ever having held it.
To reshape a payoff. Give up the top to protect the bottom, or the reverse. This is what a structured product is doing underneath its name.

The four families, and there really are only four

A forward — we agree today a price for a later date, and we both have to go through with it. The plainest one: an FX forward.
A future — the same bargain, standardised and traded on an exchange with a clearing house between the two sides and margin posted daily. Equity future, commodity future.
A swap — a series of exchanges rather than one: I pay you fixed, you pay me floating, for five years. Interest rate swap is the archetype.
An option — the only one that is not symmetric. One side has a right and the other an obligation, and the right is paid for up front. What an option is.

Everything else with a long name is one of these four, or two of them stapled together. A cap is a strip of options. A swaption is an option on a swap. A total return swap is a swap whose floating leg is a share price.

The number in the headline is not the risk

This is the single most misread fact about derivatives, and it is misread in print constantly. A contract has a notional — the reference amount its payments are calculated on. A swap with a notional of 100 million never moves 100 million; it moves the difference between two rates on that amount, which is a fraction of a percent of it. The notional is the yardstick, not the exposure.

What is at risk is two separate things: how far the underlying can move against you, and whether the person on the other side will still be there to pay. Those are different questions with different answers, and the second one is the one that gets forgotten in calm markets.

Why leverage arrives whether you asked for it or not

A forward costs nothing to enter. A future costs a margin deposit that is a small fraction of the exposure. An option costs a premium that is a small fraction of what it controls. In every case a small amount of money commands a large amount of movement — and that is leverage arriving as a property of the contract rather than as a decision somebody made.

It is why margin exists, why a margin call arrives at the worst possible moment by construction, and why you can lose more than you put in with some of these and not with others.

Two questions that unpack any derivative

Who pays whom, when, and under what condition? Every product page here that needs one carries a diagram answering exactly that — parties as boxes, payments as labelled arrows. If you can name every arrow, you understand the contract.
What is the worst case for each side, and is it bounded? A bought option's worst case is the premium. A sold option's may not be bounded at all. Two sides of one contract, two completely different shapes of risk.

Where it has gone wrong

1993 — a hedge that was economically defensible and a margin timetable that was not.
1995 — futures, and one person holding two roles that must never be one.
2021 — total return swaps, and the same position at several banks with none of them seeing the whole.

The one sentence to take away

A derivative is a promise about what something else will do, so the two things worth knowing about any of them are exactly who owes what under which condition and whether either side's worst case has a floor.