A morphism of schemes $f:X \to Y $ is called separated if the image of the diagonal morphism $\Delta: X \to X \times _Y X$ is closed.
I'm not able to grasp the picture behind this definition. Usually the diagonal being closed means Hausdorff, but schemes are generally not Hausdorff, since the Zariski topology is not Hausdorff. So what's the idea behind a proper morphism? some separation condition? is there some geometric or even algebraic intuition I should keep in mind?
You seem to already know that if $X$ is a topological space, then $X$ is separated iff the diagonal $\Delta: X \longrightarrow X \times X$ is closed. The fact is, in algebraic geometry, schemes are almost never Hausdorff so we have to find a way to translate the previous assertion in topology to algebro-geometric counterpart. We note that a topological space is nothing than a morphism $X \longrightarrow \bullet$ ($\bullet$: one-point space), this is the relative point of view, that is spaces are spaces over some base space, and the product $X \times X$ is just $X \times_{\bullet} X$. Therefore if you are working with scheme over $Y$, it is reasonable to speak of $X \times_Y X$.
Now let's turn to proper morphisms. In topology, a continuous map $f: X \longrightarrow Y$ is called proper iff the inverse image of a compact subset is again compact. But I would like to say that sheaves theory does not work very well with general topological spaces but instead one should restrict to the class of locally compact, Hausdorff spaces. In that case, being proper is equivalent to being universally closed (base change along any $Z \longrightarrow Y$ induces a closed map).
To have in mind some intuition, you should try to understand the (separatedness and properness) valuative criterions. Hartshorne and wikipedia are sufficient. Roughly, the one for separated morphisms assert that separated schemes do not contain a line with double origins (the most well-known example of a non-Hausdorff space).