Definition of elliptic curve, smoothness is needed? Why is a genus 1 curve smooth and is it still true for a non-zero genus one in general?
According to this answer, Elliptic curve（which is defined as genus 1 curve with base point）
need not to be smooth. They are just binational to the smooth 3-dimmensional curve.
Are there any example of elliptic curve which is not smooth? Thank you.

It depends where you are coming from. Every non-smooth curve is birational to a smooth curve of the same genus. Hence if a curve $C$ has genus $1$ and has a smooth rational point, you can just blow up the singular points. A concrete example of such a family are (the projective closure of)
$$y^2 = f(x)$$
where $f(x)$ is of degree $4$, has no repeated roots, and $O = (a, 0)$ is a rational point (where $f(a) = 0$). In this example the point at infinity $(0:1:0)$ is singular.
