Can regression replace correlation? Could one explain the value of correlation in statistics?
If regression is a so advanced technique, why would we even need correlation? Could we also use theoretically use regression for measuring a degree of relationship between two variables? What is so special about correlation? Just a convenient, standardized statistic for showing degree of relationship between two variables?

Pearson correlation is
$$
r_{x,y} = \frac{s_{x, y}}{s_x s_y}
$$
where $s_x$, $s_y$ are standard deviations and $s_{x, y}$ is covariance. You can estimate the parameter of simple linear regression as
$$
\hat\beta =  \frac{s_{x, y}}{s_x^2} = r_{x,y} \frac{s_y}{s_x}
$$
Correlation is the standardized regression coefficient so that it ranges from -1 to 1. It's not really about one being better, more advanced, or replacing the other. They are just differently scaled, so in some scenarios, you could prefer one and in some the other. You can also just use the covariance that appears in both.
