Hedging is usually used for two purposes...
i) to secure a rate or price in the future - e.g. I know I need to buy some Gold bullion in december for my Jewellery businesss, I can hedge the risk with a Gold future and be certain that I won't have to pay more than $830 per oz.
ii) in a strictly trading role, hedges rarely take out ALL of the risk - as Arb mentioned, a "perfect" hedge would effectively leave you flat the market, except with more spread and commissions than closing your original position completely.
More often, hedging is used to take out some of the risk, not all. An example of this might be buying Bund futures and selling 10yr futures simultaneously - the idea is that the two instruments are highly, but nor perfectly, correlated. If I think that Bonds are going down, but the Bund it too valatile for me on it's own, I can take out some of the "volatility risk" by taking an oppossite position in something that should behave in a similar fashion. As long as the Bund goes down more than the 10yrs (as long as I make more on my short Bund trade than I lose on my Long 10yr trade), I'm in profit. This is known as an inter-commodity spread, google it along with "hedge ratio's" for some examples. You can do this with all sorts of things; stocks, commodities, bonds, etc...
You can do a similar thing with multiple expiries of the same futures contract - as a general rule, contracts that expire further in the future are more volatile than the nearer dated ones - so in the example above, I could sell December Bund and hedge it with September Bund futures - if the Bund goes down as I suspect, I can expect the December future to fall more than the September one, so my profits from the Dec future should cover my losses on the Sep future, and leave me a little profit (an "Intra-contract" spread). As the two are higly correlated, this is a much less volatile position that the September future on its own.
In options trading, hedging is a popular way to reduce a type of risk completely. Option prices are affected by more than just the price of the underlying (e.g. Gold, or the Bund). They are also affected by volatility, time, all sorts. Say an options trader though that an Oil option contract was pricing volatility too high; he might want to sell the volatility part of the option, but not expose himself to the risk that the price of Oil will have on his position. By using a variety of instruments, he can create a portfolio that hedges out the "price of oil risk", leaving him just with "volatility risk", which is what he has taken a view on. This is known as "Delta hedging", again plenty around on the web.
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A final point (misnomer) is the role of hedging in the currency markets. The way that currencies are traded means that the hedging described in point ii) doesn't exist. If you are long GBP/USD, and decide to hedge your position with EUR/USD, you actually end up with a GBP / EUR position - the long and the short USD trades cancel each other out completely. So, be careful if you think you are hedging a volatile pair, say EUR / JPY, because infact you will end up exposed to a position that you didn't buy or sell in it's own right.
The exception to the above is inter-commodity hedges, and securing a price. If I am going to get some winter sun over Christmas in the USA, I might decide that I will hedge the uncertainty of the exchange rate, and buy my Dollars now, then I can do the sums before I go to know how much I have to spend on the shopping.
The other exeptions would be to hedge something other than a currency trade with a currency trade - for example, I might decide that I am Bearish on Oil, but it's too volatile, so I hedge my position by going long EUR / USD - the two have been very highly correlated recently, so as long as Oil goes down more that EUR / USD goes down, I am OK. Be wary of trades like this though, as these relationships do not last forever; if the correlation between EUR / USD and Oil drops, my hedge becomes less efficient, and I am left exposed to additional, rather than reduced, volatility.