A contour plot is a 2-dimensional way to visualize a 3-dimensional surface. It is done by by plotting constant z slices, called contours (a.k.a. level sets). That is, given a value for \(z\), points/lines/curves are drawn for the \((x,y)\) coordinates where that \(z\) value occurs.
This is the contour plot for the function \(z = x^2 + y^2\) near \((0,0)\). Noticing the closed contour lines with lower values in the center and higher values on the outside.
This is the contour plot for the function \(z = 4 - x^2 - y^2\) near \((0,0)\). Noticing the closed contour lines with higher values in the center and lower values on the outside.
This is the contour plot for the function \(z = x^2 - y^2\) near \((0,0)\) which is a saddle point.