The big picture regarding products and coproducts in categories I am new to category theory. I have been reading about products and coproducts in Chapter 0 of Topology: A Categorical Approach by Bradley, Bryson and Terilla.
Is it fair to say that products and coproducts are really just methods for putting two (or more) objects from a category together to create a new object within the same category?
I am trying to grasp the essence of what's going on before worrying about all the details (Cartesian product, direct product, direct sum, free product, disjoint union, etc.). I appreciate any help.
 A: You should probably try to think in terms of concrete examples to understand these notions. For example, the product of $X,Y$ in

*

*$\mathbf{Set}$ is the cartesian product $X \times Y$,

*$\mathbf{Vect}_k$ is the product of vector spaces $X \times Y$,

*$\mathbf{Grp}$ is the product of groups $X \times Y$,

*$\mathbf{Top}$ is the cartesian product $X \times Y$ with the product topology

and the coproduct of $X,Y$ in

*

*$\mathbf{Set}$ is the disjoint union $X \sqcup Y$,

*$\mathbf{Vect}_k$ is the direct sum of vector spaces $X \oplus Y$,

*$\mathbf{Top}$ is the disjoint union $X \sqcup Y$.

You should fill in the examples in posets as Qiaochu Yuan suggests, as this is again an instructive example.
In some sense, when you first think about products of objects, you should think of a construction that behaves similarly to the examples above - i.e. something like a cartesian product. A slightly more sophisticated viewpoint is that maps into products and maps out of coproducts are easy. This is a consequence of their universal properties.
