In my splintered readings, I have come to understand that algebraic varieties/ideals and their investigations/extensions/unifications dominated a large part of 20th century mathematics. Many profound results were achieved and prized awards given out for discoveries in this field.
But as an engineer it still escapes me why they are the quantities of central interest and how they fit into the big picture (i.e. how they provide connections between different fields of mathematics, which I think is why they are so widely investigated?). In my case I think the main issue is the vast terminology that one has to internalize before one can begin to understand even basic results. A Wikipedia reading inevitably turns to multi-hour link-fest.
To be more precise, my interest as an engineer arises specifically in their connection to dynamical systems theory. An algebraic approach to dynamical systems has been sporadically attempted since the 60s; and I think was initiated by Kalman in his study of dynamical systems over rings. Recently, far more general approaches have also been adopted incorporating category theory, sheaves etc. For example here and here.
So I am trying to find a coherent picture of their importance to (1) modern mathematics, (2) ODE's and differential geometry (3) systems theory.
Understandably, the question might be too wide to answer in a single answer, so multiple answers are invited. As for an idea of what I am looking for see this Quora answer. The answer beautifully breaks down what is probably a one line rejoinder for a mathematician into something even a sufficiently advanced high school student can understand (but the answer doesn't have to be that simple or long, intuition is more important).
I am not opposed to taking courses in Abstract Algebra to fully understand them, but from an engineering stand-point, a motivation for that type of commitment is hard to bring about unless I first get a big-picture idea of why/if it will be useful.