Hedge fund is a term used for an investment fund trading in securities that differs from traditional mutual funds. There is no strict definition of what a hedge fund is, more than that they are not as governed by the Securities and Exchange Commission (SEC) as most funds are. It is the simplest form of an investment partnership. Generally they are high-risk and use every trick there is to make a buck. The usual minimum investment is $1,000,000, but the more high profile players like George Soros require a lot more. The manager of the fund usually take 20% of the net profits, so it'd better be damn profitable to be worth while lending your one million dollars to it! 

The following are techniques and instruments used by hedge funds that are limited or forbidden for ordinary funds - in order to protect the investors. 

  • Shorting stocks - Borrowing stocks for a set period of time -at a commission -, selling them and then buying them back after the time has passed. Used when you think a stock will lose in value. Downside is that there is no limit in how much you can lose, if the market goes up. That means No Limit as in A Lot More Than You Invested. 
  • Borrowing money / Leverage - To use credit or borrowed means to increase one's speculative capacity. This is strictly forbidden for mutual funds. The idea is of course that you borrow a dollar, make a 50 cent profit and return $1.05, keeping 45 cents. Problem is when you lose borrowed money, it's hard to pay back...
  • Trading with options and futures - A way of betting on the future development of stocks and other securities. As in shorting, the downside in some scenarios is infinite loss, not only your initial investment. See Nick Leeson for a what not to do with stock options. 
  • Arbitrage - To buy and sell securities/assets on different markets under the assumption that the valuation of said assets are unequal between the markets. 
    Example
    : A glitch in the exchange rates market makes it possible to buy €1 for $0.91 in New York, and then sell it in London for $0.92. 
    Theoretically, if commissions can be held low, an arbitrage can generate an infinite amount of money. In reality, they don't exist as simple as in my example, but rather as a speculative difference in valuation between different types of assets. Hedge funds can use Convertible Arbitrage, which means that you at a fix price acquire long convertibles which will exchange for a set number of (usually) common shares. Then you short the underlying stocks in such a way, that regardless of the stocks future value at time of conversion, you'll make money on either the longs or the shorts. Problem is that large market fluctuations will destroy this scheme, with large losses following. 

Hedge funds are generally much more profitable than mutual funds, and also usually much less unprofitable. This is because they are allowed to protect the downside of the investment more creatively than a mutual fund. But the risks are higher, and this is partly why they cater to people with a lot of money, who can afford losing it all.

There's a lot of "statistical proof" of how much better hedge funds are compared to mutual funds or other investment types, but I have no way of validating that kind of "information" in a qualitative way, so I'll spare you. Sources include various hedge funds homepages.