There are some different definitions for fine sheaves.
Let X is a paracompact Hausdorff space, a sheaf F over X is a fine sheaf, if
a) Hom(F,F) is soft
b) For every two disjoint closed subsets A,B$\subset$X, A$\cap$B=$\emptyset$ , there is an endomorphism of the sheaf F$\to$ F which restricts to the identity in a neighborhood of A and to the 0 endomorphism in a neighborhood of B.
c) For every open cover {U$_i$} of X, there is a partition of unity 1= $\sum$ f$_i$ (where the sum is locally finite) subordinate to this covering.
The definition a) appears in the Bredon's book Sheaf theory and you can see the definitions b) and c) at nLab here.
Can you show they are equivalent?
Here is the unanswered same question.