Let $f:X\longrightarrow \mathbf{P}^1$ be a finite morphism of schemes. If $f$ is etale and $X$ is connected, we can show that $f$ is an isomorphism. That is, the projective line is simply connected. I would like to know if something more general can be said about $f$ if it is not assumed to be etale.
In this generality, it seems quite hard. But what if we assume $X$ to be also the projective line for example? Is it true that $f$ is then of the form $[x:y]\mapsto [x^n:y^n]$ in some sense?
No because $f$ can be ramified pretty much anywhere. Just think of a random rational function $f=p(x)/q(x)$ with $p,q$ coprime polynomials, $p$ non-constant and $q$ non-zero. That gives a finite morphism from the projective line to itself that is in general much more complicated than the map you suggest.
The question seems a bit vague, so let me also add one in addition to answers by Maharana, Kevin Buzzard and Qing Liu.
One simple but the most useful result for this kind of situation is the Riemann-Hurwitz formula, that relates the genus of X, genus of P^1 and the ramification points. See Hartshorne's chapter 4, or probably R. Narasimhan's book on Compact Riemann surfaces.
I encourage Ariyan to look at the general Riemann-Hurwitz formula for any nonconstant holomorphic maps from X to Y, where X and Y are both compact Riemann surfaces, and look at various consequences of this theorem.
