I try to learn more algebraic number theory and I am confused about the terminology "infinite prime" and "finite prime" in global fields.
If $K$ is a global field, do we mean by a finite prime of $K$ the same as a non-Archimedean absolute value? In this case, when $K = F(X)$ is a global function field, also the absolute value induced by the degree of polynomials would be considered finite. But some authors call this the "place at infinity". Is this something different than an "infinite place/prime"?
Can you please clarify this for me? Thank you in advance for any help!
Yes, that's right.
Yes, they're analogous but not the same. For a function field the "point at infinity" is the same kind of point as all the others, e.g. there are automorphisms of the projective line $\mathbb{P}^1$ acting transitively on all points. So calling it the "point at infinity" is an artifact of working with a specific set of coordinates, and via a different choice of coordinates any point can be regarded as "the point at infinity." So yes, somewhat confusingly, in the function field case all places are "finite" in the sense of being non-Archimedean.
Over number fields the situation changes drastically: there are Archimedean places and no automorphism exchanges them with the non-Archimedean places. These places are regarded as "completing" or "compactifying" the non-Archimedean places in the same way that "the" point at infinity of $\mathbb{P}^1$ completes / compactifies the affine line $\mathbb{A}^1$; you can see, for example, this old answer for one manifestation of this. This analogy is why they're called the "infinite places."