What is Galois theory for schemes? I have heard about "Galois theory for schemes" in this note.
I haven't read it yet and I know Galois theory and a litle bit about schemes (as what it is, some properties like seperated, irreducible, connected...) but I can't imagine how these two topics combine with each other.
Could you please explain for me what is Galois theory for schemes and what is it role in modern mathematics.
Thanks.
 A: Galois theory of schemes studies finite étale morphisms. This is the first step to étale cohomology, which is a vast and extremely rich area of mathematics with many applications.
The "main theorem" of Galois theory for schemes classifies the finite etale coverings of a connected scheme $X$ in terms of its fundamental group $π(X)$.
For a connected scheme $X$ there exists a profinite group $π_1(X)$ – the fundamental group of 
$X$, which is uniquely determined up to isomorphism, such that the category of finite étale coverings  is equivalent to the category of finite permutation representations of $π_1(X)$. 
A full discussion is given in SGA1 (So read SGA1, including the introduction).
The profinite group $π_1(X)$ is also called the étale fundamental group of the connected scheme X. 
A further topic, highly interesting for you, then will be Grothendieck's anabelian geometry, which also is very important for Mochizuki's papers on the abc-conjecture. For an explanation and the role of the abc-conjecture in modern mathematics, fortunately, many people have written "survey articles" recently. 
