For example the set of all points on a circle in $\mathbb R^2$ can be expressed by an equation. Similarly square, rectangle, parabola, interior of a circle, triangular regions, etc.
Likewise, can any subset of $\mathbb R^2$ can be expressed by a system (finite or infinite number) of equations or by inequalities ?
On
If you allow uncountably many formulas, then as Henning Makholm points out, it can be done. If you only allow finite many formulas (or equivalently, a single formula by concatenating all of them), then, interpreting your question as
"Can every subset of $\mathbb R^2$ be defined by a formula in which you allow reals as parameters?
then it is one of those 'typical' set-theoretic questions which can't be answered using our axioms of set theory¹. We cannot prove that there exists such an "undefinable" set, and we cannot prove that no such set exists¹.
¹ This is strictly speaking not correct, as it relies on the consistency of so-called measurable cardinals, which again cannot be proven. However, it seems to me that the majority of set theorists believe this to be true.
Sure. Take the indicator function of the subset = 1. Then, the subset is the set of points such that that equation is true.