Importance of Karl-Pearson Correlation We know that, if $(X,Y)$ is jointly Gaussian, then uncorrelatedness of $X$ and $Y$ implies independence of $X$ and $Y$. We also know that Karl-Pearson correlation coefficient measures only linear relationship (if any) between $X$ and $Y$. This means that if there is no linear relationship, then there is no relationship. Is this the reason Karl-Pearson correlation coefficient considered important?
 A: The Pearson correlation coefficient does describe the linear correlation between two datasets. You got this one right. It is also a part of each and every covariance matrix, as $\rho_{xy}=\sigma_{xy}/{\sigma_{x}\sigma_{y}}$. There are some other kinds of correlation coefficients, such as Spearman's rank correlation and Kendall's $\tau$.

This means that if there is no linear relationship, then there is no
relationship

Pardon my French, but this is totally wrong. variables can be dependent, can even be a function of each other - but with no correlation. Does this mean there is no relationship?
Consider the following variables: Let $\theta\sim U(0,2\pi)$ and denote the RVs $X=sin(\theta), Y=cos(\theta)$. The correlation coefficient is 0 (really easy to prove, try it!), but does that mean they are not related? Recalling the Pythagorean trigonometric identity $sin^2(\theta)+cos^2(\theta)=1$, we can formulate $X$ as a function of $Y$: $X=\pm\sqrt{1-Y^2}$. Clearly, $X,Y$ are dependent. The only meaning of zero correlation is that we cannot describe on of the as a linear function of the other, i.e there are no $a,b$ for which $X=aY+b$ for all values of $\theta$. That's it.
