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.