Arbitrage is the making of a gain through trading without committing any money and without taking a risk of losing money. The term is also used more loosely to cover a range of activities, such as statistical arbitrage, risk arbitrage, and uncovered interest arbitrage, that are not true arbitrage (because they are risky).
Many of these strategies bear some similarities to true arbitrage, in that they are market neutral attempts to identify and exploit (usually short lived) anomalies in pricing. The terminology used usually adds a qualifier to make it clear that it is not real arbitrage. The discussion below is of true arbitrage.
An arbitrage opportunity exists if it is possible to make a gain that is guaranteed to be at least equal to the risk free rate of return, with a chance of making a greater gain. This is equivalent to the definition of an arbitrage opportunity as the possibility of a riskless gain with a zero cost portfolio, because a portfolio that is guaranteed to make a profit can be bought with borrowed money.
Less rigorously, an arbitrage opportunity is a "free lunch", that allows investors to make a gain for no risk. Being less rigorous means that it is not really possible to distinguish between arbitrage and the closely related concepts of dominant trading strategies and the law of one price.
Arbitrage should not be possible as, if an arbitrage opportunity exists, then market forces should eliminate it. Taking a simple example, if it is possible to buy a security in one market and sell it at a higher price in another market, then no-one would buy it at the more expensive price, and no one would sell it at the cheaper price. The prices in the two markets would converge.
Arbitrage between markets is the simplest type of arbitrage. More complex strategies such as arbitraging the price of a security against a portfolio that replicates its cash flows. These range from the relatively simple, such as delta and gamma hedges, to extremely complex strategies based on quantitative models.
Much of financial theory (and therefore most methods for valuing securities) are ultimately built on the assumption that securities will trade at prices that make arbitrage impossible. In particular, if there is no arbitrage then a risk neutral pricing measure exists and vice versa. Although this result is not something that is used by most investors, it is of great importance in the theory of financial economics.
Although arbitrage opportunities do exist in real markets, they are usually very small and quickly eliminated, therefore the no arbitrage assumption is a reasonable one to build financial theory on.
When persistent arbitrage opportunities do exist it means that there is something badly wrong with financial markets. For example, there is evidence that during the dotcom boom the value of internet related tracker stocks and listed subsidiaries was not consistent with the market value of parent companies: an arbitrage opportunity existed and persisted.