Let $F$ be a presheaf over a topological space $X$ with values in a category $\A$ for which sheaves are defined. The sheafification of $F$ if it exists, is a sheaf $F'$ over $X$ together with a morphism $\theta: F \lra F'$ satisfying the following universal property:

For any sheaf $G$ over $X$ and any morphism of presheaves $\phi: F \lra G$ over $X$ there exists a unique morphism of sheaves $\psi: F' \lra G$ such that the diagram $$ \xymatrix{ F \ar[r]^{\theta} \ar@/_1pc/[rr]_{\phi} & F' \ar[r]^{\psi} & G } $$ commutes.

1. $s(p) \in F_p$ for all $p \in U$
2. For all $p \in U$ there is a neighborhood $V \subset U$ of $p$ and a section $t \in F(V)$ such that, for all $q \in V$ the induced element $t_q \in F_q$ equals $s(q)$

The following quote, taken from [1], is perhaps the best explanation of sheafification to be found anywhere:

$F'$ is ``the best possible sheaf you can get from $F$ '. It is easy to imagine how to get it: first identify things which have the same restrictions, and then add in all the things which can be patched together.

Bibliography

1

David Mumford, The Red Book of Varieties and Schemes, Second Expanded Edition, Springer-Verlag, 1999 (LNM 1358)