Let $f: X \to Y$ be a finite morphism of curves. Hartshorne gives the following definition of what it meas for the morphism $f$ to be ramified at $P$.
Let $Q=f(P)$ and $t \in \mathcal{O}_Q$ be the local parameter at $Q$, i.e since $\mathcal{O}_Q$ is a DVR, $\mathcal{O}_Q$ is a DVR $\frak{m} = (t) \in \mathcal{O}_Q$.
We can then view $t$ as an element of $\mathcal{O}_P$ via the natural map $f^{\#}: \mathcal{O}_Q \to \mathcal{O}_P$. The ramification index is then defines to be $$e_p=v_P(t)$$ where $v_p$ is the valuation associated to the DVR $\mathcal{O}_P$. We say that $f$ is ramified at $P$ if $v_p(t) > 1$. Ok.
From a figure below, I would guess that, without using any fancy/rigorours language, a ramified point is one at which the number of elements in a fiber abruptly changes.
Is my interpretation correct and how would I relate Hartshorne's definition connected to this?
In an intuitive sense the number of point on each fibre is constant BUT, you must count the index, so more accurately the sum of the indices is equal to the degree of the map. Thus if $e_P>0$ this must take away from some other points. A good example to study is $z\mapsto z^n$ in $\mathbb{C}$. Here $t=z^n$ and $e=n$. At all other fibers $e=1$, and there are $n$ points.