I've already answered this question elsewhere.
But I just wanted to warn you. I remember when I first learned category theory from MacLane's "Category theory for working mathematicians" I was fascinated by this object-free definition of category, especially because it shows more easily why monoids are one object-categories.
By the way I've learn later that this is not the best way to presents categories mostly because this kind of definition made it a little artificial presenting many classical examples: structured sets and morphisms between them, but also preorder and poset, in my personal opinion, are more easly understood as categories through the classical definition.
Reading the link that Zhen Lin posted in a comment, it seems to me that this kind of definition it's also more complex to generalize to the $\infty$-dimensional version (but maybe that's just me).
Edit: I also think this post can be interesting for this discussion.