From my understanding the correlation coefficient is r (the correlation between two variables) and ranges from -1,1 to tell the strength of a linear relationship. and is estimated as: $$r_{xy}=\frac{COV_{xy}}{\sigma_x\sigma_y}$$
The regression coefficient is simply the beta in a regression analysis. It tells the strength of the relationship. and is calculated through finding the slope of the line.
So...how is the covariance related to the regression coefficient?
A correlation coefficient is otherwise a normalized covariance so that the range is $[-1,1]$. That is, $\rho=\frac{\sigma_{xy}}{\sigma_x \sigma_y}$, where $\rho$ is the population correlation coefficient, $\sigma_{xy}$ is the covariance, and $\sigma_x \sigma_y$ is the normalization term, the product of the population standard deviations of $x$ and $y$.
For a sample correlation the same principle of normalized covariance applies and simplifies (i.e., the number of samples, $n$, scaling terms cancel) to $r_{xy} =\frac{\sum\limits_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})} {\sqrt{\sum\limits_{i=1}^n (x_i-\bar{x})^2 \sum\limits_{i=1}^n (y_i-\bar{y})^2}}$.
The regression coefficient of a simple linear regression is an estimator for $\frac{cov(X,Y)}{var X}$, where $cov(X,Y)$ is the covariance.
Basically, given two random variables $X$ and $Y$ is can be shown that the 'best' linear function of the random variable $X$ (not necessarily the Normal errors model) that can stand for/approximate/track $Y$ is:
$$L(X) := \mathbb{E}(Y) + \frac{\text{cov}(X,Y)}{\text{var}(X)}(X - \mathbb{E}(X))$$
