A variety over an algebraically closed field $k$ is nonsingular at a point $x$ if the local ring $\mathcal{O}_x$ is a regular local ring. Equivalently, if around the point one has an open affine neighborhood wherein the variety is cut out by certain polynomials $F_1, \ldots, F_n$ of $m$ variables $x_1, \ldots, x_m$ then it is nonsingular at $x$ if the Jacobian has maximal rank at that point. Otherwise, $x$ is a singular point.

A variety is nonsingular if it is nonsingular at each point.

Over the real or complex numbers, nonsingularity corresponds to ``smoothness'': at nonsingular points, varieties are locally real or complex manifolds (this is simply the implicit function theorem). Singular points generally have ``corners'' or self intersections. Typical examples are the curves $x^2=y^3$ which has a cusp at $(0,0)$ and is nonsingular everywhere else, and $x^2(x+1)=y^2$ which has a self-intersection at $(0,0)$ and is nonsingular everywhere else.