Mori cone and birational geometry Let $X$ be a projective and smooth algebraic variety (maybe here the hypotheses may be relaxed). If I understand correctly, Mori cone is defined as the closure of the cone in $N_1(X)$ of effective curves. I have heard several times people saying that this cone 
governs the birational geometry of $X$. 
I cannot understand the spirit of this claim.. I tried to look at some books (I was suggested as a good reference Lazarsfeld's book Positivity in Algebraic Geometry, but I can find it nowhere) but every time it just mentions something without explaining or I get lost inside all the technicalities and details and I still haven't found a reference which can give me the idea of this sentence.
Can anyone point me out a reference or just an example (exercise) with which I can mess a bit in order to clear my mind about that?
Thank you very much.
N.B.: I guess it is better if I can build my own intuition on it, therefore I should prefer an hint, i.e. exercise-like hint, rather than a complete answer. Thank you again.
 A: Let me try to say something as promised. 
First, an obvious point: slogans like this can never give the full picture. If you take this slogan completely literally, it would imply that if you knew the cone of curves, then you would know everything about the birational geometry of $X$. That would include, say, the whole birational automorphism group of $X$.
Exercise: Test this idea on $X=\mathbf P^n$. Let me know how it goes! ;)

OK, so if this is not literally true, what do people really mean when they say this? One point of view is based on the following:
Idea: Contractions $X \rightarrow Y$ from $X$ to other projective varieties correspond (up to a certain equivalence relation) with some faces of the cone of curves.
Note that I didn't say birational contractions here: in fact it is true that all morphisms out of $X$ are governed (in some sense) by the face structure of the cone of curves.
Exercise: Try to figure out what the equivalence relation is that I am talking about above, and which faces correspond to contractions. (Hint: it might be easier to think instead about the dual object, namely the nef cone. The key idea is pulling back ample line bundles.) 
This idea is generally not emphasised so much in the standard sources, which tend instead to jump into more advanced and technical things (like the Cone and Contraction Theorems). That is because it does not have so much content; still, I think it is helpful to have this clear in one's mind before getting into deeper issues.

As Takumi Murayama mentioned in the comments, the Cone and Contraction Theorems give a much more precise version of the idea above: given any extremal ray of the cone of curves on which the canonical bundle is negative, there is a corresponding morphism which contracts exactly those curves whose numerical class lies in that ray. This is a crucial idea in minimal model theory: here the cone of curves is telling you exactly how to change your variety to make it closer to being minimal (i.e. having nef canonical bundle). In this situation, then, the cone of curves really is "governing" the birational geometry". 

Of course if you really want to get a feel for these ideas, there is no substitute for working out some concrete examples.
Exercise: Compute the cone of curves of all the del Pezzo surfaces. (If you get stuck the answers are for example in Manin's book Cubic Forms, and presumably numerous other places.)
