When defining a sheaf of $O_X $-modules, or sheaves in general, I have nearly always seen it given as a functor from the category of all the open sets to another category with the usual properties.
However when looking at some specific examples, (e.g here http://arxiv.org/pdf/math/9906037v1.pdf)
We seem to define it on some open cover of the space.
Does this define the sheaf everywhere else automatically? Or is there something else going on? Please can someone try and explain this to me.
Yes, it is enough to define a sheaf on some open cover. It is a fact that it will define the sheaf completely. For example, you can see Corollary I-14 in Eizenbud-Harris "Geometry of schemes". This book has very nice explanation of the issue you are asking about.