The first hedge funds benefited (or thought they benefited) from the Black-Scholes formula used to calculate the value of options; supposedly a hedge fund manager could design an immensely complex portfolio consisting mainly of explosively volatile instruments , whose pieces were supposed to absorb each other's risk.
Hedge funds mainly avoided the consequences of the financial meltdown they helped create, racking up gains through the '00's that far exceeded the rest of the stock market.
Hedge funds supply market liquidity for the most exotic of instruments.
Originally hedge funds were based on the concept of risk hedging; high-yield investments are always riskier than low-yield ones, so a fund manager could presumably put all the money in one instrument with enormous risk and hope for the best. That is, to put it bluntly, insane. So the manager uses a strategy of hedging risk as cheaply as possible, such as a very elaborate combination of derivatives that rise in value if the main asset declines in value.
Hedge funds are organized to be very exclusive, requiring a very long commitment and limited membership. The managers are much more daring and will take much more aggressive risks than mutual funds.
During the first decade of the '00's, hedge funds outperformed most other asset classes. But when they melt down, like LTCM in 1997, it can be a huge event.