The name derives from the fact that they originally arose in connection with trying to determine the arc length of an ellipse. See:

https://people.math.rochester.edu/faculty/doug/mypapers/wayn...

They're curves (one-dimensional), not surfaces. An example of an elliptic curve is y^2 = x^3 + 1. The polynomial P(x,y) = x^3 + 1 - y^2 has degree 3. A surface is a 2 dimensional geometric shape.

Just to be clear, an ellipse is a quadratic curve. Ellipses are not elliptic curves. (They are still curves, though, as long as you restrict to plugging in real numbers, not complex.) The terminology is unfortunate.