I'm trying to understand the relationship between Weil divisors and Cartier divisors, and I would like to see why these are the same in the simple case where $X$ is a nonsingular projective curve over an algebraically closed field. What is the most concrete way of explaining the equivalence between the two sorts of divisors in this situation? In particular, if $P \in X$ is a closed point thought of as a prime Weil divisor, what is the Cartier divisor corresponding to $P$? What about the invertible sheaf corresponding to $P$?
1) The Cartier divisor corresponding to the Weil divisor $1.P$ is given by the pair $s=\lbrace (U,z),(V, 1) \rbrace$ described as follows:
2) The invertible sheaf corresponding to $P$ is the sheaf denoted by $\mathcal O(P)$.