How are elliptic curves related to ellipses?

Question

What is the eponymous relationship between ellipses and elliptic curves?

Answer 1

They have to do with Elliptic integrals, not much to do with ellipses other than the fact that these integrals are used to find the perimeter of an ellipse. See the paper "Why Ellipses Are Not Elliptic Curves".

Answer 2

Studying the arc length of parts of an ellipse led to certain complicated integrals called, naturally, elliptic integrals. It was eventually realized that inverses of these integrals had much nicer properties than the original arc length integrals, just as arc length integrals along part of a circle lead to inverse trig functions ($\arcsin x$) and we know the inverses of the inverse trig functions are much more interesting. Inverses of elliptic integrals, from the viewpoint of complex analysis, are doubly periodic and together with their derivative parametrize certain algebraic curves that are now called elliptic curves, but these curves are in no way the original ellipses that gave rise to all of this.

So elliptic curves are several steps removed from ellipses: ellipses lead to arc length integrals (elliptic integrals), which lead to elliptic functions through inversion, which lead to elliptic curves as the curves defined by algebraic equations linking elliptic functions and their derivatives.