A market is called efficient if it always reflects all available information. The following proposition can be derived from the Efficient Market Hypothesis (EMH): Asset prices, for instance stock prices, do not follow a systematic pattern over time, but instead follow a random walk; i.e. direction and size of any change in the share price is random and can therefore not be predicted from past share prices.
Or, if you look for a more mathematical definition: Suppose that y[t] is the price of a stock at time t and that y*[t] is the expected value of y[t] and that both series are integrated of order 1. The efficient market hypothesis requires that y[t] - y*[t] to be integrated of order 0, that is y[t] and y*[t] must be cointegrated.
Generally it can be considered that the assumptions of the EMH are fulfilled in large and competitive stock markets: relevant information is freely available, demand and supply is based on rational expectations. However, it can also be argued that the EMH is not valid even in well organized competitive stock markets: It seems that in bull markets price increases in one period lead to self-fulfilling expectations that prices will rise in the next period: which means that price changes are serially dependent and the EMH does not hold.
