Two or more non-stationary, integrated variables are cointegrated if there exists a linear combination of those variables which is integrated of a lower order, e.g. stationary.
Two or more non-stationary, integrated variables are cointegrated if there exists a linear combination of those variables which is integrated of a lower order, e.g. stationary. This implies that there is some equilibrium relationship between these variables.
Formally, for a $k\times 1$ vector of $I(d)$ variables $x_t$ with $d=1,2,\dots$, cointegration requires that there exists a vector $\beta$ such that $\beta^\top x_t$ is $I(d')$ with $d'<d$. For $\beta^i$, $i = 1,...,r$, $x_t$ is cointegrated with cointegrating rank $r$ and $\beta^i$s are called cointegrating vectors. A relevant special case is when $x_t$s are all $I(1)$ (e.g. random walks) and $\beta^\top x_t$ is $I(0)$.
Note that usually $\beta$ is normalized by restricting one of its elements by setting it equal to one. Also variables must be integrated of the same order, e.g. an $I(1)$ and an $I(2)$ variable cannot be cointegrated. However, between a set of three (or more) variables which are not integrated of the same order there may be a linear combination of the first two variables that cointegrates with the third.
