The first time I got in touch with the abstract notion of a sheaf on a topological space $X$, I thought of it as something which assigns to an open set $U$ of $X$ something like the ring of continuous functions $\hom(U,\mathbb{R})$. People said that sections of a sheaf $F$, i.e. elements of $F(?)$, are something which allow 'glueing' like in the example: if two functions $f:U\to\mathbb{R}$ and $g:V\to\mathbb{R}$ coincide on the intersection $U\cap V$ there is an unique function $U\cup V\to\mathbb{R}$ restricting to $f$ and $g$. So a sheaf consists of 'glueable' objects.

A presheaf, say of rings, on a topological space $X$ is a functor $F:Op(X)^{op}\to Rng$ where $Op(X)$ denotes the category of open sets of $X$. One may generalize all this using the terms 'site' and 'topos' but let's consider this easy situation. A sheaf is a presheaf fulfilling an extra condition, so there is an inclusion of categories $$ Pre(X)\leftarrow Shv(X):i. $$ Please excuse the awful notation but this inclusion functor admits a left-adjoint, the sheafification functor $$ f:Pre(X)\leftrightarrow Shv(X):i. $$

Since I got in touch with schemes, I think of a presheaf or a sheaf as of a space. There is a notation of a 'stalk' $F_x\in Rng$ at a point $x$ of $X$. I think of a stalk as the point $x$ of the space $F$. The inclusion functor and the sheafification functor both respect the stalks.

For example the sheafification of a constant presheaf is a locally constant sheaf.

My first question is:

How shoud I really think about sheafification?

A presheaf of sets is the canonical co-completion of a category: You take a (small) category $S$ which does not allow glueing (=has not all colimits) and then $Pre(S)$ has all colimits. $S$ is fully and faithfully embedded into $Pre(S)$ with the Yoneda embedding $Y:S\to Pre(S)$. This functor does not respect colimits, so, loosely speaking, the way of glueing is not respected in this transition. Maybe, considering sheaves instead of presheaves is a way of repairing this failure.

My second question is:

With respect to the interpretation above, what really makes the difference between a presheaf and a sheaf and how should I visualize that difference, if I think of a presheaf as if it is a space?

Thank you.