An elliptic curve is a plot that is represented using this math equation

y^2 = x^3 + bx + c

there are be two representations, a, disjointed elliptic cure or a continous elliptic curve.

What makes elliptic curves interesting are the math properties that are applied to them.

Elliptic curves with finite fields and modulos operators give many interesting properties and if you add to points, the third point would be connecting a line between two points and mirroring it against the x-axis.