level set
On a topographic map, a contour line connects all the points at the same altitude. A level set is exactly that idea for any function: collect every input where the function takes one fixed output value. In two variables you get level curves (contours); in three, level surfaces. They slice the input space into sheets of constant value.
Formally, the level set of f: R^n -> R at value c is the set of all x in the domain with f(x) = c. Different choices of c give different, disjoint level sets that together fill the domain. The collection of all level sets is a way to visualize a scalar field without leaving its input space — encoding the function's behavior by where its values are constant.
Two facts tie level sets to the rest of multivariable analysis. The gradient of f is perpendicular to the level set through each point (where the gradient is nonzero), because moving within a level set keeps f constant. And by the implicit function theorem, at points where the gradient does not vanish, the level set is locally a smooth surface of dimension n - 1. Where the gradient is zero the level set can have singularities — corners, crossings, isolated points.
For f(x, y) = x^2 + y^2, the level set at c = 1 is the unit circle; at c = 0 it is the single point (0, 0); at c < 0 it is empty. For g(x, y) = x^2 - y^2, the level set at c = 0 is the pair of crossing lines y = x and y = -x — a singular level set, since the gradient vanishes at the origin.
Smooth level sets where the gradient is nonzero; singular where it vanishes.