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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 ?

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    Since you say you allow an _infinite_ number of inequalities, would you accept a description of the form $$ \{ x \mid (x-a)^2>0 \land (x-b)^2>0 \land (x-c)^2> 0 \land \cdots \} $$ where $a,b,c,\ldots$ are all the points that are _not_ in your desired set? (Even if there are uncountably many of them?) – 2017-01-21
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    @Henning Makholm if $x, a , b, c, \ldots \in \mathbb R^2$, then how to find $(x-a)^2, (x-b)^2, (x-c)^2, \ldots$ ? – 2017-01-21
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    x @Fractal: For vectors we could say $(\left< x_1,x_2\right>-\left)^2 = (x_1-a_1)^2 + (x_2-a_2)^2$. – 2017-01-21

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Sure. Take the indicator function of the subset = 1. Then, the subset is the set of points such that that equation is true.

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    Is it the only way ? – 2017-01-21
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    This answers the question as written, but it may be worth noting that it becomes circular if "*expressed*" is replaced with "*defined*" in the question. – 2017-01-21
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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.

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    The approximate distance between this answer, the question, and the OP is that measurable cardinal that you mentioned... – 2017-01-22
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    Yeah, this is why I shouldn't answer questions late at night.. – 2017-01-22
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    Should be more appropriate now. – 2017-01-22