I am reading the Mac Lane"s book: Categories for the Working Mathematician now,but I do not understand the "graph" in it.What is the different from the graph in the category and from the module theory?How does the graph in the category generate the free category?Please give me some suggestions.
I don't know the graph in module theory, so I cannot answer your question there. A category has a lot of structure, namely
It has objects it has arrows it has composition. It has identities
A graph is a category where you don't take into account the last two things on this list. So you just have a list of objects and some arrows between them.
You can formally add compositions of these arrows, and add the identities. You will get a category in this way. This is the free category.
A graph as Mac Lane uses it, is also called a 'multidigraph (with loops allowed)', or 'quiver' in graph theory. Every category has (by definition) such an underlying graph: the objects are the vertices, the arrows (morphisms) are the directed arrows, and for every vertex there is a loop, namely the identity morphism.
I believe the term quiver is mainly used in representation theory, and this is probably the thing you mean by the term 'graph' in module theory.
Also see the graph page at the nLab.