Pearson correlation between a variable and its square Here is my R code to get familiarised with Pearson's correlation. I generate values of $X$ from 1 to 100, then find the correlation between $X$ and $X^2$:
x=1:100
y=x
for(i in 1:100) {y[i]=x[i]*x[i]}
cor.test(x,y, type="pearson")

I get this result :
Pearson's product-moment correlation

data:  x and y
t = 38.668, df = 98, p-value < 2.2e-16
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.9538354 0.9789069
sample estimates:
      cor 
0.9687564 

$r$ seems high to me.
My question is: what exactly does the $r$ coefficient quantify? Does it only quantify the closeness of the relationship between $X$ and $Y$ variable to a linear relationship ?
Or is it also suited to quantify the intensity of a relationship between $X$ and $Y$ broadly speaking (whether this relationship is close to linearity or not)?
My last question is: are there other correlation test better suited than Pearson's test to quantify the intensity of the relationship between two given variables when the kind (linear, quadratic, exponential, etc.) of this relationship is not known a priori or is Pearson's test sufficient to do this kind of job?
 A: The Pearson correlation measures the closeness to a linear relationship. If $X$ is positive, then the correlation between $X$ and $X^2$ is often fairly close to 1. 
If you want to measure the strength of monotonic relationship, there are a number of other choices, of which the two best known are the Kendall correlation (Kendall's tau), and the Spearman correlation (Spearman's rho)
 x=1:100
 cor(x,x^2,method="pearson")
[1] 0.9688545
 cor(x,x^2,method="kendall")
[1] 1
 cor(x,x^2,method="spearman")
[1] 1

I'd add that looking at the correlation of non-random values isn't necessarily where I'd start - it can be useful when exploring edge cases, however.
For the Pearson correlation you may find it useful to consider playing about with the rho and n values here:
n=100
rho=0.6
x=rnorm(100)
z=rnorm(100)
y=rho*x + sqrt(1-rho^2)*z
plot(x,y)
cor(x,y)

(In particular, you might try varying rho from close to -1 up to close to 1)
You may also find these discussions of correlation useful for getting a handle on what correlations do and don't do:
Why zero correlation does not necessarily imply independence
Does the correlation coefficient, r, for linear association always exist?
If A and B are correlated with C, why are A and B not necessarily correlated?
How would you explain covariance to someone who understands only the mean?
Pearson's or Spearman's correlation with non-normal data
How to choose between Pearson and Spearman correlation?
Kendall Tau or Spearman's rho?
If linear regression is related to Pearson's correlation, are there any regression techniques related to Kendall's and Spearman's correlations?
