In finance, this is an investment strategy whereby the investor will profit is the price of the asset falls. This is in contrast to a "long position" where the investor is betting on the price going up.
The most common way this is done is to borrow the "physical" asset and sell it. For example, if one is going to short a particular stock, he or she would borrow it, most likely from a brokerage firm. The selling of the security means the money is in the trader's account. For this to be profitable, the price of the stock needs to fall, providing an opportunity to buy at a lower price. The reason for this is the asset needs to be returned to the brokerage firm. By getting it at a lower price, the "returned" asset is worth less than the one that was initially received and sold.
Another common for of shorting is to use derivatives. This is not technically short selling since nothing was sold. One is betting the value of the derivative moves up, hence it is still a long play. The difference is that asset is designed to move in opposite direction of the underlying security. In options, a put is an example of this.