Visualizing an implicitly defined surface
This is a plot of the surface z=(sin(x)+cos(y))/(1+x^2+y^2)

ContourPlot3D[(Sin[x]+Cos[y])/(1+x^2+y^2)-z==0,{x,-6,6},{y,-6,6},{z,-.5,1.75},AxesLabel->{x,y,z}]

We can pass a plane z = const through this and look at the curves we get. These are called traces :

Animate[ContourPlot[(Sin[x]+Cos[y])/(1+x^2+y^2)-z==0,{x,-6,6},{y,-6,6}, PlotPoints->40,PlotRange->{-6,6}],{z,-.5,1.25,.0025}, AnimationRate->0.25,AnimationRunning->False,AnimationRepetitions->1, AnimationDirection->Forward]

We can also draw all of those contours at once :

ContourPlot[(Sin[x]+Cos[y])/(1+x^2+y^2),{x,-6,6},{y,-6,6},Contours->60,PlotPoints->60,ClippingStyle->Pink,MaxRecursion->Automatic]

Here are the plots of the "standard models,"  and their traces:
First: x^2+y^2+z^2=1

ContourPlot[x^2+y^2-1,{x,-2,2},{y,-2,2},PlotPoints->40, ClippingStyle->Pink, Contours->Function[-Range[-2,0,.05]^2]]

ContourPlot3D[x^2+y^2+z^2==1,{x,-2,2},{y,-2,2},{z,-2,2}]

Next : -x^2 + y^2 + z^2 = 1

ContourPlot[-x^2+y^2-1,{x,-2,2},{y,-2,2},PlotPoints->40, ClippingStyle->Pink, Contours->Function[-Range[-4,0,.25]^2]]

ContourPlot3D[-x^2+y^2+z^2==1,{x,-2,2},{y,-2,2},{z,-2,2}]

Next : -x^2 + y^2 + z^2 = -1

ContourPlot[-x^2+y^2+1,{x,-2,2},{y,-2,2},PlotPoints->40, ClippingStyle->Pink, Contours->Function[-Range[-4,0,.25]^2]]

ContourPlot3D[-x^2+y^2+z^2==-1,{x,-2,2},{y,-2,2},{z,-2,2}]

Next : x^2 + y^2 - z = 0

ContourPlot[x^2+y^2,{x,-2,2},{y,-2,2},PlotPoints->40, ClippingStyle->Pink, Contours->Function[Range[0,4,.25]]]

ContourPlot3D[z-x^2-y^2==0, {x,-2,2}, {y,-2,2}, {z,0,2}]

Next : x^2 - y^2 - z = 0

ContourPlot3D[z-x^2+y^2==0, {x,-2,2}, {y,-2,2}, {z,-2,2}]

ContourPlot[-x^2+y^2,{x,-2,2},{y,-2,2},PlotPoints->40, ClippingStyle->Pink, Contours->Function[Range[-2,2,.25]]]

Here' s a sphere (orange) and a more general ellipsoid (blue) :

ContourPlot3D[{x^2+y^2+z^2==1,9*x^2+4*y^2+z^2==3},{x,-1.75,1.75}, {y,-1.75,1.75}, {z,-1.75,1.75},Mesh->None, ContourStyle->{Directive[Orange,Opacity[0.8],Specularity[White,30]],Directive[Blue,Opacity[0.8],Specularity[White,30]]},PlotPoints->30]