Hedging is the act of reducing one’s financial exposure to an event or price movement. Insurance is considered a classic form of hedging.
Consider an investment bank that has taken on a large short position on Stock A. They may then buy an out-of-the-money call option on that stock, so that if the stock price massively rises they can recoup some of their losses if the stock gains in value.
Event contracts are a newer form of hedging. An event contract is a kind of contract that pays out if a given event occurs. Suppose one is a small businessman who will lose $10,000 if an income tax proposal becomes law, and the bill has a 50% chance of becoming law. If they buy 10,000 contracts that pay out $1 if the bill passes (for 50 cents each), they then lose $5,000 if the bill become laws (since they lose $10,000 from the bill plus $5,000 from buying the contract, but offset that with the $10,000 payout from the event occurring), and $5,000 if the bill does not (they only lose the $5,000 from buying the contract). In short, hedging transforms an uncertain gain/loss into a certain (but smaller) gain or loss.