Arbitrage is a strategy involving the simultaneous purchase and sale of identical or equivalent financial instruments across two or more markets in order to benefit from a discrepancy in their price relationship (e.g., equivalent tools in the cash and futures markets).
In a theoretical "efficient market," there is a lack of opportunity for profitable arbitrage. An efficient market is one in which market prices adjust rapidly to reflect new information. The degree to which the market is efficient depends on the quality of information reflected in market prices. In an efficient market, profitable arbitrage opportunities do not exist and traders cannot expect to consistently outperform the market unless they have lower-cost access to information that is reflected in market prices or unless they have access to information before it is reflected in market prices.
An arbitrageur is an individual or institution that engages in a form of arbitrage.
Merger arbitrage would be that of, for example, in October of 2006 when Chicago Mercantile Exchange (NYSE: CME) proposed a merger with the Chicago Board of Trade (NYSE: BOT). For every 10 shares of CBOT that investors held, the terms of the merger would have given them slightly more than three shares of CME. CBOT's share price rose sharply on the announcement.
To use merger arbitrage for a potential profit, an investor could have bought 10 shares of CBOT and sold short three shares of CME. The short sale would have given the investor a total of $1,536 in proceeds, while buying CBOT shares would only have cost $1,510. If the merger went through, the investor would receive three CME shares, which could be used to cover an earlier short sale.
Statistical arbitrage involves an attempt to profit from pricing inefficiencies that are identified through the use of mathematical models. Statistical arbitrage attempts to profit from the likelihood that prices will trend toward a historical norm. 
Fixed-income arbitrage is an investment strategy which consists of the discovery and exploitation of inefficiencies in the pricing of bonds.
Convertible arbitrage is a strategy involving the purchase of convertible securities and the subsequent shorting of the corresponding stock. Conversion will offset the short position. The transaction may be profitable if the convertible is priced incorrectly relative to the stock.