How to render a 3D ellipsoid with Graphics3D?

Using Sphere with Scale and Rotate works too:

Graphics3D[Rotate[Scale[Sphere[], {5, 4, 2}, {0, 0, 0}], 60 Degree, {1, 2, 1}]]

The first triple is the scaling in the x,y,and z coordinates, the second triple is the translation, and the third triple is the axis about which to rotate. To generate a number of random ellipses:

x := RandomReal[];
Show[Table[Graphics3D[Rotate[Scale[Sphere[], {x, x, x}, {x i/6, x i/6, x i/6}], 
                      x, {x, x, x}], Boxed -> False], {i, 25}]]

You can modify this if you need to specify the rotation in different ways, etc. As Simon Woods has suggested, probably the best way is to use GeometricTransformation.

 ellipsoid[a_, b_, center_?VectorQ, rotation_, around_?VectorQ] := Fold[
           GeometricTransformation,
           Sphere[],
           {ScalingTransform[{a, b, b}],
            RotationTransform[rotation, around],
            TranslationTransform[center]}]

 ellipsoid @@@ Table[{x, x, 10 {x, x, x}, x, {x, x, x}} /. x :> RandomReal[]
                     , {111}] // Graphics3D // AbsoluteTiming

{0.347020,

An alternative approach that generates explicit primitives instead of transformed ones uses the NURBS representation of a sphere, with all the appropriate transformations done to its control points to generate the ellipsoid:

myEllipsoid[dims : {_?Positive, _?Positive, _?Positive} : {1, 1, 1}, 
            center : (_?VectorQ) : {0, 0, 0}, 
            rot : {_, _?VectorQ} : {0, {1, 0, 0}}] := Block[{ctrlpts},
  ctrlpts = Composition[TranslationTransform[center], 
                        RotationTransform[Sequence @@ rot],
                        ScalingTransform[dims]] /@ 
            Outer[Append[#2 #1[[1]], #1[[2]]] &,
                  {{0, -1}, {1, -1}, {1, 1}, {0, 1}},
                  {{1, 0}, {1, 1}, {-1, 1}, {-1, 0}, {-1, -1}, {1, -1}, {1, 0}}, 1];
            BSplineSurface[ctrlpts, SplineClosed -> True, SplineDegree -> 2, 
                           SplineKnots -> {{0, 0, 0, 1/2, 1, 1, 1},
                                           {0, 0, 0, 1/4, 1/2, 1/2, 3/4, 1, 1, 1}}, 
                           SplineWeights -> Outer[Times, {1, 1/2, 1/2, 1},
                                                  {1, 1/2, 1/2, 1, 1/2, 1/2, 1}]]]

Here's an example:

randomEllipsoid := myEllipsoid[RandomReal[1, 3], RandomReal[{-2, 2}, 3],
                               {RandomReal[{-π, π}], 
                                Normalize[RandomVariate[NormalDistribution[], 3]]}]

BlockRandom[SeedRandom[42, Method -> "Legacy"]; 
            Graphics3D[Table[{ColorData[61, RandomInteger[{1, 9}]], randomEllipsoid},
                             {50}], Boxed -> False, Lighting -> "Neutral"]]