Functions of two variables

Here is a simple function of two variables

F[x_, y_] = x*y

Evaluated at a few points:

F[0, 0]
F[1, 2]
F[3, 3]
F[-1, 0]

The following plots our graph for values of x and y lying in the square: -1<x<1 and -1<y<1:

Plot3D[F[x, y], {x, -1, 1}, {y, -1, 1}]

The BoxRatios->Automatic option makes each axis appear to be at the same scale

Plot3D[F[x, y], {x, -1, 1}, {y, -1, 1}, 
      BoxRatios -> Automatic]

We can use the ViewPoint option to change our point of view:

Plot3D[F[x, y], {x, -1, 1}, {y, -1, 1}, 
      BoxRatios -> Automatic, 
      ViewPoint -> {2, 3,  .125}]

Here

For[i = 0, i < 2*Pi, i += Pi/20, 
      Plot3D[F[x, y], {x, -1, 1}, {y, -1, 1}, 
        BoxRatios -> Automatic, 
        ViewPoint -> {3 Cos[i], 3 Sin[i],  .125}, 
        SphericalRegion -> True]]

ContourPlot[F[x, y], {x, -1, 1}, {y, -1, 1}]

The Coutours option allows us to plot more level sets

ContourPlot[F[x, y], {x, -1, 1}, {y, -1, 1}, 
      PlotPoints -> 130, Contours -> 31]

Here we color the contours with colors

ContourPlot[F[x, y], {x, -1, 1}, {y, -1, 1}, 
      PlotPoints -> 130, Contours -> 31, 
      ColorFunction -> Hue]

Here's an example where we render some pretty pictures with Mathematica

F[x_, y_] := 1/(Cos[x]*Cos[y] + 1.2)

The larger we make PlotPoints, the smoother the surface will appear, but the longer it will take to draw
Mesh->False takes away the black grid lines from our surface

Plot3D[F[x, y], {x, -6, 6}, {y, -6, 6}, 
      BoxRatios -> Automatic, PlotPoints -> 200, 
      PlotRange -> All, Mesh -> False]

PlotPoints->100 makes the level sets appear smoother

ContourPlot[F[x, y], {x, -6, 6}, {y, -6, 6}, 
      PlotPoints -> 100, ColorFunction -> Hue]

We can specify which contour lines to plot with the Contours Option
In the folowing plot, we choose level sets correspoinding to F[x,y]=.5, .75, and 1

ContourPlot[F[x, y], {x, -6, 6}, {y, -6, 6}, 
      PlotPoints -> 100, ColorFunction -> Hue, 
      Contours -> { .5,  .75, 1}]

Contours can also be used to specify the number of contour lines to plot as below.
We also have increased PlotPoints which makes the level curves more smooth.
ContourLines->False removes the black level set lines from the image.

ContourPlot[F[x, y], {x, -6, 6}, {y, -6, 6}, 
      PlotPoints -> 400, ColorFunction -> Hue, 
      ContourLines -> False, Contours -> 200]