Here is my current understanding of what a minimal polynomial is:
$p(x)$ is the minimal polynomial of $a$ over a field $\mathbb{F}$ if $p(a) = 0$ and $p(x)$ is the lowest degree irreducible, monic polynomial in $\mathbb{F}[x]$ for which this is true.
Is this exactly correct? Please tell me if not.
Now my question is, what are minimal polynomials used for? What applications does it have in algebra research? Does it tell us anything about the field we are working in?
For a correct definition of minimal polynomial within field theory see here. So we start with a field extension $L/K$. You did not specify where $a$ is from, and did not mention a field extension.
It is widely used in field theory, but also in linear algebra (for a specific definition see here). For many proofs in abstract algebra it is quite useful. It is furthermore used for many results in algebraic number theory (involving for example norm and trace of field extensions, rings of integers, and discriminants).