Why construct new groups from the old ones? I am self-studying Hungerford's book Algebra. In the page $59$ he wrote the following:

In this section we study products in the category of groups and
  coproducts in the category of abelian groups. These products and
  coproducts are important not only as a means of constructing new
  groups from old, but also for describing the structure of certain
  groups in terms of particular subgroups (whose structure, for
  instance, may already be known).

I would like to know why is it important to construct new groups from the old ones? 
 A: In applications, you might run into some group that you care about (e.g. it might act as automorphisms on some object that you care about). How do you describe and study this group? Well, probably a good idea is to describe in terms of other groups you understand and use your understanding of those groups to understand this new group. 
For example, sometimes you'll run across a finitely-generated abelian group (e.g. because of Dirichlet's unit theorem or the Mordell-Weil theorem). That's great because the structure theorem tells you that all such groups are finite products of cyclic groups, so you can understand them by understanding cyclic groups. 
As another example, the modular group $\text{PSL}_2(\mathbb{Z})$ is a group of great importance to the theory of modular forms and related subjects. Surprisingly, it can be described as the coproduct $\mathbb{Z}/2\mathbb{Z} * \mathbb{Z}/3\mathbb{Z}$. 
A more general class of important examples is computing fundamental groups using the Seifert-van Kampen theorem. 
You'll run into more examples like this the more you keep studying mathematics, so my advice is not to seek a definitive list of examples or anything like that. Just trust in the general principle; it is very fruitful. 
A: One more, cliched but fundamental, example of why it's important to know how groups can be constructed from other groups.
As you know, finite simple groups have been classified into 16 infinite families and an additional 26 groups that don't fit into these families. ( http://en.wikipedia.org/wiki/List_of_finite_simple_groups )
A finite group theorist I know describes much of what he does as "tabular mathematics:"
Step 1) Prove that some statement holds for all the finite simple groups. (Here's where he has to go through the table of all such groups, hence "tabular."
Step 2) Prove that if $G\cong H/K$ and the statement holds for $H$ and $K$, then it holds for $G$.
Needless to say, this is a very fruitful approach, and it is more or less the main approach to any program for proving the inverse Galois problem.
