What's the relationship between cointegration and linear regression? If two non-stationary processes are cointegrated, that means a linear combination of the two processes are stationary.  In a simple linear regression, we have the model form:
$y = b_0 + b_1x + e$
If we re-arrange, we can have something like
$(y - b_1x) = b_0 + e$
And thus, the linear combination of y and x are stationary with mean b0 and variance $\sigma^2$. If y and x are stock prices, then $b_1$ is the hedge ratio.
So what are the similarities and differences of cointegration and simple linear regression?  I am not seeing the big picture for cointegration yet and why it is useful.  The typical example of cointegration has to do with stock prices.  Why not just take any two stocks prices, run a linear regression between them, check the residuals and make sure it passes the typical SLR assumptions?  Basically the residuals show stationarity.  And thus we can use typical regression methods as opposed to an entirely new suite of cointegration tests and methods.
 A: Cointegration and regression are quite different categories.
Cointegration is a phenomenon observed in a time series context. Several time series cointegrate if there exists a linear combination that is integrated of a lower order than the series themselves. (See also the tag description for cointegration.)
Regression has several meanings. The most relevant is perhaps the one in the tag description of regression which says it is techniques for analyzing the relationship between one (or more) "dependent" variables and "independent" variables.
The relationship between cointegration and regression is that one can use regression to analyze the relationship between several cointegrated variables.
(Unlike the simple case of cross-sectional data, standard regression estimators such as OLS of a naive regression of several cointegrating variables have some unusual properties, e.g. superconsistency. A helpful regression model for cointegrating time series is cointegrated (restricted) VAR and its alternative representation VECM that clearly exposes the short- and long-run relationships between the variables.)
A: On the level of data generating processes, cointegration is a special case of linear regression. (In this sense, I disagree somewhat with @RichardHardy.)
Say the time series $(x_t, y_t)$, $t = 1, 2, \cdots$, follow a linear regression if
$$
y_t = \beta x_t + \epsilon_t, \mbox{ where } E[\epsilon_t] = 0.
$$
If we agree on this terminology, that clearly a cointegrating relationship is a special case of linear regression. You might call it a "cointegration regression".
The difference is distributional assumptions on data generating process $(x_t, y_t)$, $t=1,2,\cdots$.
In a usual regression model. $(x_t, y_t)$ is stationary. For cointegration, $x_t$ and $y_t$ are both non-stationary but the linear combination $y_t - \beta x_t$ is.
These two settings are very different, from both statistical and empirical perspectives. (In this sense, I don't disagree with @RichardHardy.)
For example, statistically, under stationarity, OLS $\hat{\beta}$ is consistent only if $E[x_t \epsilon_t] = 0$ (or at least $\frac{1}{n} \sum_{t=1}^n E[x_t \epsilon_t] \rightarrow 0$).  Under cointegration, $\hat{\beta}$ is always super-consistent.
Empirically, cointegration is about modelling long-run equilibrium relationships whereas under stationarity the regression describes a contemporaneous relationship.
A: Regression and cointegration, in a nutshell, are different  things. A cointegration relation comes out from a VECM approach. In this way is just a long term relation inside a system with short term elements(VECM SYSTEM). Naturally if you have just two variables cointegrated, regression equation will be also the contegration relation. They are equal. But in more dimension, three or more, the equations will be different and you should use appropriate methods.
