What are the implications of knowing the algrebaic structure(group, ring, monoid, etc) of a set? I remember groups, rings, monoids, lattices, etc. being taught in my undergraduate mathematics course.
I never really understood what they were for. After that lesson, we moved on to other lessons without looking back to this specific one.
So, what exactly are they for? What are its implications for mathematicians, and what does it mean for mathematics beginners?
 A: One reason to care about mathematical structures is they allow us to prove things "once and for all." For example, we can prove that if a poset has all bounded-above non-empty joins, then it has all bounded-below non-empty meets, and vice versa. Therefore, since $\mathbb{R}$ can be viewed as a poset with respect to the usual order relation, and since the completeness axiom tells us that $\mathbb{R}$ has all bounded-above non-empty joins (i.e. suprema), hence our theorem tells us that it has all bounded-below non-empty meets (i.e. infima). We don't have to "re-prove" this for the special case of $\mathbb{R}$.
There is at least one more reason to care about mathematical structures, namely the homomorphisms. At a basic level, homomorphisms are important because they preserve lots of goodies. For example, if:


*

*$X$ and $Y$ are monoids, 

*$\varphi : X \rightarrow Y$ is a monoid homomorphism

*$x,x' \in X$ are elements


then $$xx' = e_X \;\implies\; f(x)f(x') = e_Y.$$
So monoid homomorphisms preserve inverses, and therefore, they preserve invertibility.
Taking a more sophisticated viewpoint, structures and their homomorphisms tend to form into categories. For example, there is a category $\mathbf{Grp}$ whose objects are groups and whose arrows are group homomorphisms. There is also usually a so-called forgetful functor to $\mathbf{Set}$ (see here). A major application of this occurs in abstract algebra, whereby we can describe the construction of free algebras as left-adjoint to the forgetful functor. This leads to the idea of presentations; e.g. we can present a group by generators and relations.
