The definition of divisor I know is: Let $X$ be a Riemann surface, a divisor is a map $D\colon X \to \mathbb{Z}$ such that for every compact $K \subset X$ only finitely many points $x\in X$ have $D(x) \neq 0$. (From Forster's Lectures on Riemann Surfaces).
I do not understand what $D = nP$ is supposed to mean for $n \in \mathbb{N}$, $P$ a point. What precisely is the definition of nP? What is the geometric intuition behind this?
A comment on my background: I know classical algebraic geometry and compact Riemann surfaces as far as Forster's book goes.