How can a non-mathematician intuitively understand the importance of algebraic varieties?

Solution 1:

Classically, an algebraic variety is defined as the set of solutions of a system of finitely many polynomial equations over the real or complex numbers. The polynomials $f(x_1,\ldots ,x_n)$ are in $n$ variables. How can we solve polynomial equations exactly, and not just numerically? How does the solution set look like? Are there algorithms?

Algebraic Geometry and algebraic varieties in generality are of course more than this, and a survey of topics, links, and references shows why algebraic varieties are important, e.g., here:

Why study Algebraic Geometry?

Help understanding Algebraic Geometry