Meaning of correlation In probability books it is sometimes mentioned that correlation is a measure of linearity of a relationship between random variables. 
This claim is supported by the observation that $\left| \rho(X,Y)\right|=1 \iff X=b+aY$.
But let consider a less extreme case: $X,Y,Z$ are random variables such that $\rho(X,Y)=0.5$ and $\rho(Y,Z)=0.8$. Does it mean that the relationship between $X$ and $Y$ is "less" linear than between $Y$ and $Z$? How at all the degree of linearity in a relationship can be defined (especially given the fact, that a relationship between random variables can be non-functional, so one can not always use the second derivative)?    
Edit:
I am talking about Pearson correlation here.

Perhaps rather than saying "less linear" you might say "less well modeled by a line". If you think of a scatter-plot of $(x,y)$ pairs, it might be sort of a disk or filled ellipse of points, or an amorphous blob. If it's an ellipse, then you'd say that a line drawn along the major axis of the ellipse "sort of fits" the data, especially if the ellipse is highly eccentric. This is exactly the situation in which the correlation is large. If the scatter plot is amorphous, like the standard drawing of an amoeba, then you'd say that it's hard to fit any particular line to the shape; in this case, correlation tends to be small. 
I don't think that there's anything deeper here than that; the idea is simply that when the correlation is high, a scatter plot of the data tends to look pretty much like a highly eccentric ellipse, which itself looks a  lot like a line.   
