I think you'll find out very different my example of algebraic structures.

Monoid: the monoid of words over a finite alphabet.

Group: the group of words over a finite alphabet.

Ring: the polinomial ring.

Module: $\mathbb K^n$ for some $\mathbb K$ field.

Category: the path category over some graph $\mathbb G$.

$R$-Algebra (for some ring $R$): $M_n(R)$, i.e. the ring of matrix over the ring $R$.

These is exactly those algebraic structures that come in my mind when I think/try to prove fact about algebraic structures. Why? Because they're free object and are the terms models of their corresponding algebraic theories, so every other algebraic models derive from these by adding relations.