What is the relationship between covariance and correlation coefficient in the equation? I want to know about covariance and correlation coefficient.
What is their relationship in terms of equation?
What are their similarities and differences?

Let $\sigma_X$ and $\sigma_Y$ be the standard deviations of two random variables, $X$ and $Y$, respectively. Then the (Pearson) correlation between $X$ and $Y$, denoted $\rho(X,Y)$, is defined as:
$$
\rho(X,Y)=\dfrac{
cov(X,Y)
}{
\sigma_X\sigma_Y
}
$$
In words, correlation is the covariance scales by the product of the standard deviations.
Another way to think of correlation is as the covariance between two variables after the variables are scaled to have have unit variance by dividing each by its corresponding standard deviation.
$$
cov(X/\sigma_X,Y/\sigma_Y)=\dfrac{1}{\sigma_X\sigma_Y}cov(X,Y)\\=\rho(X,Y)
$$
In general, $cov(aX,Y)=a\times cov(X,Y)$ and $cov(X,bY)=b\times cov(X,Y)$, by the linearity of the expected value definition of covariance, explaining how the division by the standard deviations can be pulled out of the covariance.
$$
cov(X,Y)=\mathbb E\left[
(X-\mu_X)(Y-\mu_Y)
\right]
$$
