Stick a bit to the syntax and everything works:

ClearAll[f, a, b, c, d]
f[x_, y_] := (1 - x^2 - y^2)/(1 + x^2 + y^2) 
a = -4; b = 4; c = -5; d = 5


(* F= *)Plot3D[f[x, y], {x, a, b}, {y, c, d}, 
 PlotLabel -> "3D.Diagramm", AspectRatio -> 1/GoldenRatio, 
 AxesLabel -> {"x", "y", "z"}, PlotRange -> {{a, b}, {c, d}, {-1, 1}}]




(* Diagramm= *)ContourPlot[f[x, y], {x, a, b}, {y, c, d}, 
 PlotLabel -> "Höhendiagramm", ColorFunction -> "BlueGreenYellow", 
 AspectRatio -> 1/GoldenRatio]

the function definition was in error and the definition for a,b,and c needs a closing semicolon. d as the last one can go without a semicolon: CompoundExpression