Two time series $X_1$ and $X_2$ are cointegrated if a linear combination $aX_1+bX_2$ is stationary i.e. it has constant mean, standard deviation and autocorrelation function for some $a$ and $b$. In other words, the two series never stray very far from one another.
Cointegration might provide a more robust measure of the linkage between two financial quantities than correlation which is very unstable in practice.
I have borrowed the following two examples from Willmot's Frequently Asked Questions in Quantitative Finance, one may be typical for a hedge fund trader and another illustrates the job of a mutual fund manager.
A. Suppose you have two stocks $S_1$ and $S_2$ and you find that $S_1 − 3 S_2$ is stationary, so that this combination never strays too far from its mean. If one day this ‘spread’ is particularly large then you would have sound statistical reasons for thinking
the spread might shortly reduce, giving you a possible source of statistical arbitrage profit. This can be the basis for pairs trading.
B. Suppose we find that the S&P500 index is cointegrated with a portfolio of 15 stocks. We can then use these fifteen stocks to track the index. The error in this tracking
portfolio will have constant mean and standard deviation, so should not wander too far from its average. This is clearly easier than using all 500 stocks for the tracking (when, of
course, the tracking error would be zero).