an algebraic curve
Draw a circle and you can describe it without ever pointing at it: it is exactly the set of points (x, y) for which x^2 + y^2 = 1. The equation does the work. An algebraic curve takes this idea as far as it goes: it is the set of points in the plane that make some polynomial equation in x and y equal to zero. A line (ax + by + c = 0), a circle, a parabola (y - x^2 = 0), a figure-eight, a cusp — all of them are algebraic curves, each one the 'zero set' of a polynomial. Geometry becomes a conversation about polynomials.
More precisely, fix a polynomial f(x, y) — a finite sum of terms like 3x^2y, -5xy, or 7 — and the curve is the locus of all (x, y) with f(x, y) = 0. The DEGREE of the curve is the highest total degree of any term (x^2y has total degree 3), and it controls almost everything: degree 1 is a line, degree 2 the conics (ellipse, parabola, hyperbola), degree 3 the cubics (which include the elliptic curves behind modern cryptography). A curve is IRREDUCIBLE if its polynomial cannot be factored into two lower-degree polynomials; if f = g times h, the curve is really two curves laid on top of each other (g = 0 together with h = 0). Working over the complex numbers, and adding points at infinity in the projective plane, makes the theory clean — there the number of intersections and the degree behave with no annoying exceptions.
Algebraic curves are the doorway to algebraic geometry, one of the deepest subjects in mathematics, and they are genuinely useful: elliptic curves secure internet traffic, and curves classify by an integer called the genus that ties the algebra back to topology. One honest caution: over the real numbers a polynomial's zero set can be surprising. x^2 + y^2 = 0 has only the single point (0, 0), and x^2 + y^2 + 1 = 0 has NO real points at all, yet both are 'degree 2 curves' algebraically. The slogan 'a curve is a one-dimensional thing you can draw' is a good first picture but not the definition — the definition is the zero set, and over the reals it can shrink to a point or vanish.
Consider f(x, y) = y^2 - x^3 + x. Its zero set is a cubic curve. To plot a point, you do not solve anything fancy: pick x = 0, then y^2 = 0 so y = 0, giving (0, 0). Pick x = 1, then y^2 = 1 - 1 + 1 = 1 so y = +1 or -1, giving two points (1, 1) and (1, -1). Sweeping x across the real line and recording where y^2 stays non-negative traces out the visible curve, which here splits into an oval plus an open branch.
A curve is read off its polynomial point by point; degree 3 here allows the curve to split into separate pieces.
Degree is not the same as 'how wiggly it looks'. A high-degree polynomial can trace a gentle-looking arc, while a humble degree-2 conic can be a wide-open hyperbola; the degree counts intersections with lines, not visual complexity.