Level curves of surfaces z=f(x,y)

Plot3D[4-x^2-y^2,{x,-2,2},{y,-Sqrt[4-x^2],Sqrt[4-x^2]}]

Lines on surface above are called traces (x = constant or y = constant)

Contour

Plot3D[4-x^2-y^2,{x,-2,2},{y,-Sqrt[4-x^2],Sqrt[4-x^2]},MeshFunctions->{#3&},Mesh->5]

Or we can plot the cross-sections (z = constant)

ContourPlot[4-x^2-y^2,{x,-2,2},{y,-2,2}]

ContourPlot[{4-x^2-y^2==0,4-x^2-y^2==1,4-x^2-y^2==2,4-x^2-y^2==3,4-x^2-y^2==3.9},{x,-2,2},{y,-2,2}]

Plot3D[x^2-y^2,{x,-2,2},{y,-2,2}]

Plot3D[x^2-y^2,{x,-2,2},{y,-2,2},MeshFunctions->{#3&},Mesh->9]

ContourPlot[x^2-y^2,{x,-2,2},{y,-2,2}]

Plot3D[Sin[x]+Sin[y],{x,-2Pi,2Pi},{y,-2Pi,2Pi}]

Plot3D[Sin[x]+Sin[y],{x,-2Pi,2Pi},{y,-2Pi,2Pi},MeshFunctions->{#3&},Mesh->9]

ContourPlot[Sin[x]+Sin[y],{x,-2Pi,2Pi},{y,-2Pi,2Pi}]

Plot3D[x y Exp[-x^2-y^2],{x,-2,2},{y,-2,2}]

Plot3D[x y Exp[-x^2-y^2],{x,-2,2},{y,-2,2},MeshFunctions->{#3&},Mesh->6]

ContourPlot[x y Exp[-x^2-y^2],{x,-2,2},{y,-2,2}]