ag.algebraic geometry - degree of pull-back via F of an hyperplane vs degree of defining polynomials of F

Let X be a nice projective variety (say, smooth and defined over the complex numbers) embedded in some projective space, and let $F:Xtomathbb{P}^n$ be a rational map given by $[f_0:f_1:cdots: f_n]$ where the $f_i$ are coprime homogeneous polynomials of degree d (of course, there is just one way to choose them satisfying this, up to constants factors). Consider H the hyperplane divisor $mathbb{P}^n$, and H' the hyperplane divisor on X.
It seems that F*(H)=dH', at least in this case. I have two questions about this situation;
(1)What is the name of this number? I couldn't find it anywhere.
(2)What is the relation between this number and the degree of the map when F is finite (if any)? One may think they are the same but the Segre embedding shows that this is not the case.
Thanks in advance.